diff --git a/data/00basic/00title b/data/00basic/00title
index e3a0785b97b946b14e0ba019942aad7e7692a0ea..0bd606537b7f7bd973fb7b4fd6a7eb79044ae087 100644
--- a/data/00basic/00title
+++ b/data/00basic/00title
@@ -1 +1 @@
-<titlepage title="Deep learning for new physics mining at the LHC" subtitle="" name="Simon kluettermann" department="Institute for theoretical particle physics and cosmology" university="RWTH Aachen" country="Germany">
\ No newline at end of file
+<titlepage title="Deep learning for new physics mining at the LHC" subtitle="" name="Simon Kluettermann" department="Institute for theoretical particle physics and cosmology" university="RWTH Aachen" country="Germany">
diff --git a/data/01intro/00title b/data/01intro/00title
index 185d1865b21ce033a90894c879e7f36a15caeb22..200b82495ac9dbb8eed0bd34641358c26a395517 100644
--- a/data/01intro/00title
+++ b/data/01intro/00title
@@ -1 +1 @@
-<section title="Introduction and literature" label="secintro">
\ No newline at end of file
+<section title="Introduction and literature" label="secintro">
diff --git a/data/01intro/01intro b/data/01intro/01intro
index 69930d8e3172b410c0fac71a9c2269bc93c435d0..9ca5cace2246f3dd4a52e25cd4fccdc18a6e9f1a 100644
--- a/data/01intro/01intro
+++ b/data/01intro/01intro
@@ -39,7 +39,7 @@ Current state of this thesis
   <e>a couple notes in brackets</e>
   <e>a few images in solution 1 either not optimal or missing entirely for computational reasons</e>
   <e>some results not yet final(just numbers i found somewhere, not yet optimized, so things look a bit worse than they hopefully will)</e>
-  <e>spellchecking not done. I wrote a programm for this, but after half an hour it bugged out, and i dont have the patience to fix it now. So there are spelling errors, but you can just ignore them</e>
+  <e>spellchecking not done. I wrote a programm for this, but after half an hour it bugged out, and i don`t have the patience to fix it now. So there are spelling errors, but you can just ignore them</e>
   <e>all references missing...those i do first thing tomorrow</e>
   <e>some mmt images have wrong aspect rations</e>
 
@@ -79,7 +79,7 @@ Current state of this thesis
 <e> Images: often missing, their description is then just a description for me what image i want to put there, or not missing, then usually not the final image, since there are no axes labels etc. There description is then an exact location of the file that generatet them</e>
 <e>chapters: 1,2 reread, the rest a bit messy, especcially the chapters about old models</e>
 <e>missing chapters: 5,7 mostly, since im still working a lot on them, some minor chapters in between, and i think i could still double the length of this thesis by things i migth want to write about</e>
-<e>chapternames: still not always the best, since these were originially descriptions of what i want to write about in this chapter, and i did not always chance them</e>
+<e>chapternames: still not always the best, since these were originially descriptions of what i want to write about in this chapter, and i did not always change them</e>
 <e>Links and references: completely missing, references just because this is not a priority yet (But i know what i want to put where), and links because not every chapter exists</e>
 
 
diff --git a/data/01intro/02physics b/data/01intro/02physics
index d24859ad688fe20d7ac455a69cc1bc3306f3c400..affff12df3930e2b75643be011adaf6a13dda459 100644
--- a/data/01intro/02physics
+++ b/data/01intro/02physics
@@ -1,13 +1,13 @@
 <subsection title="New physics" label="physics">
 
-Modern particle physics seems to be in a standstill: The standart model seems to explain everything on a small size, while also beeing clearly incomplete. At the same time, each suggested extension seems to be either wrong or untestable, which is why there are now approaches changing the fundamental way we do science. One of this approaches is suggested by QCDorWhat <cite QCDorWhat>: Instead of finding new physics events by hypotising theories and testing them afterwards, use an anomaly detection algorithm to filter out events that dont match your expectation. This allows you to find events representing new physics models without needing to suggest these models first. They work here in jet physics, trying to find anomalous jets that are generated by the decay of a top quark, while only knowing about those jets, which are generated by the decay of QCD<note quantum chromodynamics, with QCD particles you usually mean low mass quarks and gluons> particles with lower mass. You can think of this task, as trying to finding new physics, while only knowing as much as we did before the detection of the top quark in 1995<cite topquark>. This suggests that when we would be able to find top quarks at this point in time, we migth also be able to apply such an algorithm to the LHC now and use it to understand physics no human knows about yet.
+Modern particle physics seems to be in a standstill: The standart model seems to explain everything on a small size, while also beeing clearly incomplete. At the same time, each suggested extension seems to be either wrong or untestable, which is why there are now approaches changing the fundamental way we do science. One of this approaches is suggested by QCDorWhat <cite QCDorWhat>: Instead of finding new physics events by hypotising theories and testing them afterwards, use an anomaly detection algorithm to filter out events that don`t match your expectation. This allows you to find events representing new physics models without needing to suggest these models first. They work here in jet physics, trying to find anomalous jets that are generated by the decay of a top quark, while only knowing about those jets, which are generated by the decay of QCD<note quantum chromodynamics, with QCD particles you usually mean low mass quarks and gluons> particles with lower mass. You can think of this task, as trying to finding new physics, while only knowing as much as we did before the detection of the top quark in 1995<cite topquark>. This suggests that when we would be able to find top quarks at this point in time, we migth also be able to apply such an algorithm to the LHC now and use it to understand physics no human knows about yet.
 
 <ignore>
-If you think about the way, modern physics works, there is one thing, that always strike me as odd: If you today look at the arxiv and simply search for a term like string theory, it returns about #25000# Paper: What is there usage? I mean it is safe to assume, that there is nobody who knows everything that is in them, while each of them still seemed at some point to have some scientific merit sure, for each paper, there are some people that know what's in them, but this still means, that theremighth be two paper, that perfectly combine into one world chancing paper, but nobody ever knew about both? Sure, youmighth say, that this is improbable, as string theory is fairly general, and everybody has his ownspecialtyy in which he knows a much higher fraction of all paper, but the problem is, that science profits a lot from applying ideas from different sides of science (Consider the symmetry breaking and the higgs or even the CMB between cosmology and radiodetection(ENTER better begriff something like nachrichtechniklike)), and then, these bubbles actually hurt your understanding, as most people who read a given paper also read the same other paper. And even if we think, somebody read everything, these are still theories, and if we simply assume that each theory has one experiment, that is run to test this theory, and for simplicity each experiment results in the same significance, then, for #25000# theories we would expect #ENTER# 4 sigma false positives, and even an #0.01# Chance for a false 5 sigma positive. This migth not seem so much, but the assumptions are also fairly weak: there are more than #25000# Theory paper, and most theories don't just result in a single test for them, and at the end, this still means, that testing more theories becomes at least a lot harder, the more theories are given.
+If you think about the way, modern physics works, there is one thing, that always strike me as odd: If you today look at the arxiv and simply search for a term like string theory, it returns about #25000# Paper: What is there usage? I mean it is safe to assume, that there is nobody who knows everything that is in them, while each of them still seemed at some point to have some scientific merit sure, for each paper, there are some people that know what's in them, but this still means, that theremighth be two paper, that perfectly combine into one world changing paper, but nobody ever knew about both? Sure, youmighth say, that this is improbable, as string theory is fairly general, and everybody has his ownspecialtyy in which he knows a much higher fraction of all paper, but the problem is, that science profits a lot from applying ideas from different sides of science (Consider the symmetry breaking and the higgs or even the CMB between cosmology and radiodetection(ENTER better begriff something like nachrichtechniklike)), and then, these bubbles actually hurt your understanding, as most people who read a given paper also read the same other paper. And even if we think, somebody read everything, these are still theories, and if we simply assume that each theory has one experiment, that is run to test this theory, and for simplicity each experiment results in the same significance, then, for #25000# theories we would expect #ENTER# 4 sigma false positives, and even an #0.01# Chance for a false 5 sigma positive. This migth not seem so much, but the assumptions are also fairly weak: there are more than #25000# Theory paper, and most theories don't just result in a single test for them, and at the end, this still means, that testing more theories becomes at least a lot harder, the more theories are given.
 
 So what can we do? Or better, what can a computer do. The obvious idea migth be to let a computer read each paper, and let it try to draw its own conclusions, but since we cannot<note usual ml gives very little importance to understandanding what their network does, which is not a good idea for serious scientific work, also it has very little respect for abstract usability: Consider gpt(ENTER REFERENCE): A fairly impressive code, that surely is able to generate texts that look like paper, and even working code (ENTER REFERENCE) for generating machine learning models, but this does not mean, that this model is truly able to understand their output: The model migth run, but gpt is not at all able to optimize it> Do this yet, we suggest another way: 
 
-Modern physics consists out out of 10th of thausends of theories<note searching for a term like supersymmetry on arxiv returns #14604# paper at the time of writing this thesis, while string theory returns #24918# and dark matter returns #34797#.>, which all migth be rigth about something, but designing an analysis or even an experiment for each of those theories would take an incromprehensible amount of time, which is why you want to take another approach: Find something first, and test which theories can explain them: The problem here is, that we basically only know two things that dont match into the standart model<note this is obviously a huge simplification, but there is still nothing safely measured at the LHC violating SM> Dark matter(/Energy) And neutrino masses. The effect of neither of those could yet be measured at the LHC, in fact, there is no real anomaly measured there, or at any other particle accelerator, at all: The whole output of the LHC does not violate the standardrt model expectation, or at least, we could not find anything. And that is weird: It is fairly wide believed that the standart model is just a low energy approximation of a true theory<note (ENTER REFERENCE)>, but this theory would have slight effects even on energies that we are already able to measure. And sure, you can create a theory, that simply does not show any difference at LHC energies, but would not you have to ask yourself what is the value of this theory, when its only reason for existing is the fact that it is not excluded already: In short, you want an anomaly, preferable at LHC energies. So how to look for this anomaly? Consider the detection of the positron (ENTER REFERENCE), which could be detected by only one particle, one single track that just could not be explained otherwise, you have to ask yourself if you cannot do this again? Find a single event in the huge mass of LHC events, that is different enough to clearly show non SM physics. So why can't we do this? Mostly, there is a lot of LHC data,beingg in the hundreds of trillions events (ENTER REFERENCE). Combine this, with the fact, that an LHC event can be much more complicated than a single antilepton track, the fact that this single track was worth a nobel prize and the stories of other physicists that could have detected the positron(ENTER REFERENCE), and this seems basically impossible. Or at least impossible for humans to do, but just maybe not for a computer.
+Modern physics consists out out of 10th of thausends of theories<note searching for a term like supersymmetry on arxiv returns #14604# paper at the time of writing this thesis, while string theory returns #24918# and dark matter returns #34797#.>, which all migth be rigth about something, but designing an analysis or even an experiment for each of those theories would take an incromprehensible amount of time, which is why you want to take another approach: Find something first, and test which theories can explain them: The problem here is, that we basically only know two things that don`t match into the standart model<note this is obviously a huge simplification, but there is still nothing safely measured at the LHC violating SM> Dark matter(/Energy) And neutrino masses. The effect of neither of those could yet be measured at the LHC, in fact, there is no real anomaly measured there, or at any other particle accelerator, at all: The whole output of the LHC does not violate the standardrt model expectation, or at least, we could not find anything. And that is weird: It is fairly wide believed that the standart model is just a low energy approximation of a true theory<note (ENTER REFERENCE)>, but this theory would have slight effects even on energies that we are already able to measure. And sure, you can create a theory, that simply does not show any difference at LHC energies, but would not you have to ask yourself what is the value of this theory, when its only reason for existing is the fact that it is not excluded already: In short, you want an anomaly, preferable at LHC energies. So how to look for this anomaly? Consider the detection of the positron (ENTER REFERENCE), which could be detected by only one particle, one single track that just could not be explained otherwise, you have to ask yourself if you cannot do this again? Find a single event in the huge mass of LHC events, that is different enough to clearly show non SM physics. So why can't we do this? Mostly, there is a lot of LHC data,beingg in the hundreds of trillions events (ENTER REFERENCE). Combine this, with the fact, that an LHC event can be much more complicated than a single antilepton track, the fact that this single track was worth a nobel prize and the stories of other physicists that could have detected the positron(ENTER REFERENCE), and this seems basically impossible. Or at least impossible for humans to do, but just maybe not for a computer.
 <ignore>, this means, that a single event, could be enough for you to prove your theory, but how to find this event in the petabytes of data generated by the LHC till now? You can simply test every single event for your theory, and compare it to the standart model prediction, but this still requires you to have a theory, so what we really want, is to do this without a theory at all:
 What you could do, is to find the theoretical expectation for each measurable, and compare it to the actual measurement. Sadly this is way easier said than done: Your first problem might be randomness: A top quark can decay in a huge amount of ways, and so, you would need to test every possible top decay chain compared to your data, and even this assumes that you at some point have a top quark: at the end, you have to compare hugh amount of possible SM predictions. This will undoubtely miss a lot of anomal events, as they are just to similar to a different chain, but even when you can manage this, there is still the problem of observables: You can easily compare two jets for example by their total momentum, but you can also define infinitely many other observables, and unless you work on a very basic level<note ideally raw detector currents, but we use here 4 momenta, as they are easier to get> You cannot be sure you don't miss anything: and when you usethe lowestt level data, this makes the task of comparing tostandardndart model much harder, possibly also resulting in you miswhat'swhats important. So how to accomplish this? And even if we are able to do this, programming this seems like a lot of work, so why not let a computer also do program this? Find a way, of letting a computer llearn what it means to be a standard model event, and looking more closely at those events, the computer deems not SM. And even though we will spend the rest of this thesis making this work, we want to stress that this is by no means trivial: Defining what is not like the standart model expects it to be, means defining the standart model, and since the only way we can reliably communicate with the network, is through the data we classify as standart model, this basically means learning as much of the standart model as possible, from just raw experiments<note or more precisely simulations>. And when you consider, that this is a task, that took some of brightest minds of the last century literally decades to do, redoing this on a computer, which has no prior knowledge, without much oversight possible and in a matter of hours, this is a task that borders on hybris.
 But when you are able to do this well? Not only would you be able to detect anything abnormal without knowing what you form this anomaly takes, or prove that there is nothing abnormal in a given dataset, but you also make a huge step on the way to learning theories without human input.<ignore> That being said, we are still a long way from this bbeingperfect, which is why we need this thesis
diff --git a/data/01intro/03ae b/data/01intro/03ae
index ee2ca3dc873f95ed678152910c4a735483740911..4e04bfb7fe1081772c1d50202a97dd2aa1c31235 100644
--- a/data/01intro/03ae
+++ b/data/01intro/03ae
@@ -9,7 +9,7 @@ Autoencoder <cite aebasic> are a special kind of these neural networks, in which
 As seen in <refi aeexamp>, to completely encode the data you would still require 2 dimensions (a #x# and a #y# value), but you can approximately encode them into 1 dimension quite well, by using one value as this compressed state and reconstruct the second one in the decoder, as a linear function of the compressed state<note since the number of trainingsamples is finite, you could map every sample into an index, and map those indices again onto the inputs. This would reach a zero loss for any input with an compression size of 1. The problem is that not only is finding such a function quite hard for a neural network, it would also not be useful at all, since on any new data (for example the validation data), the network would not work at all. This is why these kind of functions are a part of what you can call overfitting for autoencoder><note in practice, this data contains structural noise, which is why the autoencoder would not learn a linear function, but a more complicated on, better representing the data>.
 
 
-This combination of a compresser and a decompresser can be quite useful in multiple ways. Ignoring the obvious task of compressing data (see <cite aeforcompress> for an example of an autoencoder used in particle physics), you can give the decompressor noise to generate new versions of an already known kind of data (see <cite aeforgen>), and even though nowadays GANs(Generative adversial networks see <cite aeforgen> or <ref tipsy>) are used for this task, autoencoder have still some benefits, allowing for more control over the generated data. This works, since (for good autoencoders<note this works better in a special way of training an autoencoder, called a variational autoencoder <cite variationalae>.>) similarity in the compressed space represent similarity of the inputs. This does mean, that by identifiyng features in the input space you can chance just one attribute of an input, and you can also combine the features of two inputs into one, see for example <cite aecombine>
+This combination of a compresser and a decompresser can be quite useful in multiple ways. Ignoring the obvious task of compressing data (see <cite aeforcompress> for an example of an autoencoder used in particle physics), you can give the decompressor noise to generate new versions of an already known kind of data (see <cite aeforgen>), and even though nowadays GANs(Generative adversial networks see <cite aeforgen> or <ref tipsy>) are used for this task, autoencoder have still some benefits, allowing for more control over the generated data. This works, since (for good autoencoders<note this works better in a special way of training an autoencoder, called a variational autoencoder <cite variationalae>.>) similarity in the compressed space represent similarity of the inputs. This does mean, that by identifiyng features in the input space you can change just one attribute of an input, and you can also combine the features of two inputs into one, see for example <cite aecombine>
 <i f="aemerge1" f2="aemerge2" f3="aemerge3" wmode="True">Example images showing how autoencoder can combine two images into one. Taken from <cite ostagram> (Ember, Bruno, ArgonOl)</i>
 That being said, the application of autoencoders that is focussed on in this work, is the detection of anomalies. As introduced on the informatics side in <cite aeforanomaly> and a bit later for particle physics in <cite aeforanomalyphys>: Since a well trained autoencoder should be only able to reconstruct the features it is trained on well  you can use the reconstruction loss<note the difference between input and output of the autoencoder, measured in a way discussed in chapter <ref losses>> of this autoencoder, to find events that are not of the same type as the data the autoencoder is trained on. This allows QCDorWhat <cite QCDorWhat> to find top jets in a background of QCD jets with an noteable precision, without ever needing to know how a top jet looks like.
 
diff --git a/data/01intro/05graphs b/data/01intro/05graphs
index a27c9b1c38af1e23a70b1486d7c9da6b186897a5..67995bb2ab7ae2379817fd94a73e154c32121b83 100644
--- a/data/01intro/05graphs
+++ b/data/01intro/05graphs
@@ -3,7 +3,7 @@
 A graph<cite graphbasic> is a mathematical<ignore>/informatical</ignore> concept, that allows to define a more general form of data then by just encoding it in vectors. Namely, graphs allow storing relational information of an arbitrary<note for computional reasons, graphs are not completely unbounded in the following chapters, but have a maximum size, up to which their size is arbitrary.> amount of objects. This is done, by defining two objects: Nodes which are the objects of interest and can be mathematically described by vectors<note in theory you would not need to be able to define those objects as only vectors, but for practical application this is quite useful. chapter <ref thirdworking> could be interpreted as using graphs themself as the information encoded in those nodes.> and edges that are pairs of connected nodes and thus encode the relation between those objects of interest
 <note there are multiple extensions for this simple graph, the two most important ones are directed graphs, in which the edges gain a direction, and thus a connection between node #i# and #j# does not automatically imply a connection between #j# and #i# and also weighted graphs, in which each edge gains an additional value, that encodes how strong the connection between two nodes is.>.
 <i f="dia3" wmode="True" wid="0.7">(mmt/dia3) A simple graph explaining nodes and edges</i>
-Mathematically, these nodes and relations are defined in a list of feature vectors<note technically this is equivalent to a matrix, but list of vectors is more intuitive> #X# that stores the features of each node, and an adjacency matrix #A#, which components #A_i**j# are #1#, if the nodes #i# and #j# are connected, and #0# if not. This graph is usually invariant under permuation of the node indices. You achieve this by permuting<note with permuting we here mean switching two indices, or more generally multiplying with a permutation matrix> the adjacency matrix in the same way the feature vectors are permuted<note this is the reason why we dont call the list of feature vectors a matrix: as a matrix permutation requires permutation matrices on each side (#p*A*p#), the feature vector "Matrix" only requires one permutation matrix (#X*p#)>, and by requiring any action on the graph to be permuation invariant. This action is usually also local, and thus only acts on each node and the mean<note you also need to require each local action to be symmetric under chancing the input ordering, since else the output can depend on the order of the nodes. The usual way that is achieved is by using a function like the mean on all neighbours> of nodes that are connected to the current node<note this works, since permuting indices does not chance, which nodes are connected together>. This has the benefit of making graphs ideal for modeling interactions between high numbers of objects, as the functions dont chance as you add more nodes to the graph. In informatics, this is useful for example for social networks<cite graphsoc>: Data that consists out of a huge amount of nodes in which mostly only connected nodes (friends) affect each other, are perfect applications for graphs, since else you would need to update your model every time a new user joins. In physics, this reminds of nuclear science, and the approximation of pair interaction potentials<cite pairpot>, and so there are applications using this kind of molecule encoding for chemical feature extraction (for a simple example look at chapter <ref mol>) <cite gnnforchemistry> and medicine <cite gnnformedicine>.
+Mathematically, these nodes and relations are defined in a list of feature vectors<note technically this is equivalent to a matrix, but list of vectors is more intuitive> #X# that stores the features of each node, and an adjacency matrix #A#, which components #A_i**j# are #1#, if the nodes #i# and #j# are connected, and #0# if not. This graph is usually invariant under permuation of the node indices. You achieve this by permuting<note with permuting we here mean switching two indices, or more generally multiplying with a permutation matrix> the adjacency matrix in the same way the feature vectors are permuted<note this is the reason why we don`t call the list of feature vectors a matrix: as a matrix permutation requires permutation matrices on each side (#p*A*p#), the feature vector "Matrix" only requires one permutation matrix (#X*p#)>, and by requiring any action on the graph to be permuation invariant. This action is usually also local, and thus only acts on each node and the mean<note you also need to require each local action to be symmetric under changing the input ordering, since else the output can depend on the order of the nodes. The usual way that is achieved is by using a function like the mean on all neighbours> of nodes that are connected to the current node<note this works, since permuting indices does not change, which nodes are connected together>. This has the benefit of making graphs ideal for modeling interactions between high numbers of objects, as the functions don`t change as you add more nodes to the graph. In informatics, this is useful for example for social networks<cite graphsoc>: Data that consists out of a huge amount of nodes in which mostly only connected nodes (friends) affect each other, are perfect applications for graphs, since else you would need to update your model every time a new user joins. In physics, this reminds of nuclear science, and the approximation of pair interaction potentials<cite pairpot>, and so there are applications using this kind of molecule encoding for chemical feature extraction (for a simple example look at chapter <ref mol>) <cite gnnforchemistry> and medicine <cite gnnformedicine>.
 Next to those relational applications, there are also applications that are not utilizing an existing relation, but use the locality of the graph structure to encode the similarity of given data. This is done by letting the sense of similarity between nodes be a learnable function. For example, by using a topK algorithm (each node is connected to its neirest #K# neighbours, see <ref atopk>), you can implement a learnable version of whatever distance means. This allows networks like for example particleNet <cite particeNet>, which uses a special kind of neural network, that is able to work on graphs, to seperate top and QCD jets<note with QCD we mean jets that are generated by gluons or other quarks that are not top quarks> in a supervised way. They use the graph structure, to be able to define and redefine multiple times, which detected particles (nodes), should be considered close to each other. This results in particleNet beeing a quite good classifier(see <cite toptagref>).
 
 
diff --git a/data/02tech/01binclass b/data/02tech/01binclass
index 04cbb64cd8b0d62d7d8a474e2d319feb1f73c09d..001fb0c055b47f506e0a2e5295dead63d7818f7d 100644
--- a/data/02tech/01binclass
+++ b/data/02tech/01binclass
@@ -36,7 +36,7 @@ Or to focus on the fraction of falsy called signal events, which is usually call
 <i f="rocb" wmode="True">(test/rocdraw)A sample roc curve focussing on the background rejection</i>
 
 <subsubsection title="Area under the curve" label="classauc">
-To simplify comparing ROC scores, you can use an AUC (Area under the curve) score to summarise it. This AUC score is defined as the integral of the true positive rate over the false positive rate. This simplification is not perfect, since you reduce a function into only one number, but it is fairly wide accepted, as it is much easier to interpret: A perfect score would result in an AUC score of 1, while a classifier that just guesses randomly, results in an AUC score of 0.5 and a perfect anticlassifier would result in 0. <ignore>Also this reduction into only one number can make the AUC score less errorprone than other values.</ignore> On the other hand, since not every part of the roc curve is equally important for the current problem (if you want to test, if somebody is ill, you migth prefer more false positives over more hidden illnesses). This could result in networks improving the AUC score by just chancing unimportant parts of the ROC curve.
+To simplify comparing ROC scores, you can use an AUC (Area under the curve) score to summarise it. This AUC score is defined as the integral of the true positive rate over the false positive rate. This simplification is not perfect, since you reduce a function into only one number, but it is fairly wide accepted, as it is much easier to interpret: A perfect score would result in an AUC score of 1, while a classifier that just guesses randomly, results in an AUC score of 0.5 and a perfect anticlassifier would result in 0. <ignore>Also this reduction into only one number can make the AUC score less errorprone than other values.</ignore> On the other hand, since not every part of the roc curve is equally important for the current problem (if you want to test, if somebody is ill, you migth prefer more false positives over more hidden illnesses). This could result in networks improving the AUC score by just changing unimportant parts of the ROC curve.
 
 <ignore>
 
diff --git a/data/02tech/04dataprep b/data/02tech/04dataprep
index 229fbaee32e01e097f4817d53a210e22b1452d6c..ec7f67884e0680705102d30093a1b8e4b2b7aa8a 100644
--- a/data/02tech/04dataprep
+++ b/data/02tech/04dataprep
@@ -12,12 +12,12 @@ Every network has a fixed maximum number of particles that can be put into it, f
 <e>#Delta_eta#: #Eq(eta,ln((p+p_3)/(p-p_3))/2)# (with $p=|\vec{p}|$) which is shiftet in such a way, that the mean of #Delta_eta# is 0, since the position of the jet should not have any meaning: #Eq(Delta_eta,eta-mean(eta))#.</e>
 <e>#Delta_phi#: #Eq(phi,arctan2(p2,p1))#<note the function #atan2(y,x)# is an extension of #arctan(y/x)# that is able to map to the full #2*pi# output space> which is again shiftet in such a way, that the mean of #Delta_phi# is 0: #Eq(Delta_phi,phi-mean(phi))#<note here is shifting actually not that easy to implement, since you have to consider the difference in a modular space, see appendix <ref aimplementationphi> or the implementation (ENTER GIT LINK gpre5) for more information>, since also this position of the jet should not have any meaning.</e>
 
-<e>#lp_T#: #Eq(p_T**2,p_1**2+p_2**2)#, and #Eq(lp_T,-ln(p_T/p_T**jet))#. This logarithm, is needed to keep each value of about the same order of magnitude, which makes the training more stable. We also divide by the total jet transverse momentum, to make every jet look more similar (see appendix <ref alpt> for the effects of chancing this). Finally the sign is used to keep the values positive. <note a consequence is that higher transverse momenta have lower values. Since the alternative in appendix <ref alpt> changes this, we can say that this does not matter to much></e>
+<e>#lp_T#: #Eq(p_T**2,p_1**2+p_2**2)#, and #Eq(lp_T,-ln(p_T/p_T**jet))#. This logarithm, is needed to keep each value of about the same order of magnitude, which makes the training more stable. We also divide by the total jet transverse momentum, to make every jet look more similar (see appendix <ref alpt> for the effects of changing this). Finally the sign is used to keep the values positive. <note a consequence is that higher transverse momenta have lower values. Since the alternative in appendix <ref alpt> changes this, we can say that this does not matter to much></e>
 
 
 </list>
 
-You could try to use more variables: ParticleNet for example uses 4 more variables (different representations of other variables and the energy as well as #Eq(Delta_R**2,Delta_Phi**2+Delta_Eta**2)#. The also dont use flag as input), but since these variables are strongly related to other variables, this results in an autoencoder only learning those relation. This would not result in anything learned beeing usable as classifier. And demanding that this and more is learned, just complicates the task, without providing any real benefit<note but see appendix <ref afeature> for some experiments in chancing the features>.
+You could try to use more variables: ParticleNet for example uses 4 more variables (different representations of other variables and the energy as well as #Eq(Delta_R**2,Delta_Phi**2+Delta_Eta**2)#. The also don`t use flag as input), but since these variables are strongly related to other variables, this results in an autoencoder only learning those relation. This would not result in anything learned beeing usable as classifier. And demanding that this and more is learned, just complicates the task, without providing any real benefit<note but see appendix <ref afeature> for some experiments in changing the features>.
 
 
 
diff --git a/data/02tech/06explain b/data/02tech/06explain
index 1fbadc36a76e84c6e207baf04de5458de26506be..e9bc5cb9398111aae112190e9e805097fbbf256c 100644
--- a/data/02tech/06explain
+++ b/data/02tech/06explain
@@ -18,8 +18,8 @@ We show the setup of each network as images similar to those<note these images a
 <list>
 <e>The data that is trained on</e>
 <e>is getting preprocessed into 4 variables (see chapter <ref data> for more information)</e>
-<e>Afterwards it is sorted by its #lpt#<note this is done to make comparing jets easier, it is not strictly neccesary, but makes the training more effective (see appendix <ref asort), also the first sorting is really not needed, since the particles are already sorted, but applying it here means, that you dont have to take care while chancing the preprocessing of #lpt#></e>
-<e>Afterwards, each event is normated (see chapter <ref normalization> for the why and the how)</e>
+<e>Afterwards it is sorted by its #lpt#<note this is done to make comparing jets easier, it is not strictly neccesary, but makes the training more effective (see appendix <ref asort), also the first sorting is really not needed, since the particles are already sorted, but applying it here means, that you don`t have to take care while changing the preprocessing of #lpt#></e>
+<e>Afterwards, each event is normalized (see chapter <ref normalization> for the why and the how)</e>
 <e>After normalization, these are the values, we want to reconstruct, as marked by the line on the top, and another normalization, this time a standart batchNormalization, is applied. This allows the network to learn the desired scale and mean for the remaining layers(see appendix <ref abatchnorm> for why this is done)<note it is fairly important to add the batchNormalization layer after the value for comparison is choosen, since a learnable scale, would else just result in the network perfectly reconstructing zeros></e>
 <e>After the normalization, a topK layer generates a graph to the particle nodes(see appendix <ref atopkhow> for a breakdown on how this layer works)</e>
 <e>Which is used in a graph update layer (see chapter <ref gnn></e>
diff --git a/data/03graphae/01agraphnet b/data/03graphae/01agraphnet
index 55654ed364ebde4cd40d599b6a310d7e859f8182..f11a9b806be6098e12498e9f6e3a35ec85018759 100644
--- a/data/03graphae/01agraphnet
+++ b/data/03graphae/01agraphnet
@@ -5,7 +5,7 @@ Our graph update layer consists out of two matrices, a self interaction matrix,
 So written as a formula, the new vector equals (with the original feature vector #x_i#, the learnable self and neigbour matrices #s_i**j# and #n_i**j#, as well as the adjacency matrix #A_i**j# and the activation #f#)
 ##f(x_i*s_j**i+x_i*A_k**i*n_j**k)##
 (FORMULA GETS REORDERED LATER!)
-It should be noted, that this implementation has one central problem: It is a bit slower than the usual approach(for a reasoning on why we cannot use the more usual approach of particleNet, see appendix <ref acomparepnet>)<note especcially since they can utilise GPUs better>, and even though we dont think the implementation (see git (ENTER LINK)) is as fast possible, this is something that could be improved a lot <ignore>this speed deficit migth be a price to pay, for the higher number of possible actions you can apply to the graph, and thus to making graph autoencoder possible</ignore>.
+It should be noted, that this implementation has one central problem: It is a bit slower than the usual approach(for a reasoning on why we cannot use the more usual approach of particleNet, see appendix <ref acomparepnet>)<note especcially since they can utilise GPUs better>, and even though we don`t think the implementation (see git (ENTER LINK)) is as fast possible, this is something that could be improved a lot <ignore>this speed deficit migth be a price to pay, for the higher number of possible actions you can apply to the graph, and thus to making graph autoencoder possible</ignore>.
 <ignore
 finally the activation function (here labeled #f#) migth be interesting: Some inspiration in writing this layer was taking from a paper (ENTER PAPERLINK), which does graph updating a bit different compared to many other papers. Instead of having multiple different update layers, they use only one layer, that gets called multiple times, until it converges<note converging neural networks are not very easy to implement, since you cannot easily execute a function if and only of a certain condition is met, in practice converging networks are just executed a certain number of times, until you assume that anything that should converge, converged>. This seemed to be quite physical<note consider a graph describing a system of coupled harmonic oszillators, each update step migth then simulate one timestep, and a converging output would be the stable state. In fact this whole idea reminds a bit on hopfield networks, that use some clever math to set the convergent state to something fixed, and are based on ising models (ENTER REFERENCE)>, but this opens the question, on how to be sure, that your update step actually converges: This is mostly a question of the activation, since the update step itself is learnable, and thus will generally not create any matrices that have determinants to far away from 1<note i simplify here a bit, since what would actually matter, would be the combination of two different matrices, where it is not trivial to say if the result has a determinant of 1, especcially since one depends on the adjacency matrix, but the intuition migth be the same>, but even when the matrices are convergent, an activation like a sigmoid, that for each (positive) input is smaller than its input, would go to zero if applied infinite times, so we demand that the activation fullfills<note in theory, this migth not be neccesary, the matrices could equalate the activation, but since this would still demand that the network learns to equalate this, we keep this assumption>
 ##Eq(f(f(x)),f(x))##
diff --git a/data/03graphae/07eval b/data/03graphae/07eval
index 4125ad1ca65e4dbde178dbf5183d1bd8d0b6ce2f..93e62972b4afcb0fa170bddf6e72deda10ee81a4 100644
--- a/data/03graphae/07eval
+++ b/data/03graphae/07eval
@@ -4,19 +4,19 @@ Before we can look at the results of our network, we have to look at how to judg
 We migth be able to evaluate a binary classification problem<note see chapter <refs binclass>>, but evaluating an network is a bit more difficult, since we basically want to do 2 things at the same time: Creating an autoencoder and creating a classifier, so there migth be situations in which the autoencoder is good, but the classifier is bad and situations in which the classifier migth be good, but the autoencoder is basically useless.
 
 <subsubsection title="AUC scores" label="evalauc">
-If you want to evaluate a network, you migth simply use the quality of the classifier (the AUC Score, see chapter <refs classauc>): since the classifier should work by the autoencoder understanding the data, and thus should only be good if also the autoencoder is good. And in most cases this works, there is a clear relation between the quality of the autoencoder and the quality of the classifier (see chapter <ref normalization>), but in general this is simply not true, as for example chapter <ref simplicity> show. And even if your working in a region where this relation is true, Classifier evaluation methods<note AUC scores even have one of the lower uncertainities> usually have a much higher uncertainity<note uncertainity in the sense that even a well trained network can chance its AUC score by a couple of percent after retraining, even if it has the same loss (see chapter <ref auncertain>)> than other methods, which is why in the regions in which there is a strong correlation, it was more useful to use the loss of the network to assert that the network improves, and to simply know that the AUC score will correlate.
+If you want to evaluate a network, you migth simply use the quality of the classifier (the AUC Score, see chapter <refs classauc>): since the classifier should work by the autoencoder understanding the data, and thus should only be good if also the autoencoder is good. And in most cases this works, there is a clear relation between the quality of the autoencoder and the quality of the classifier (see chapter <ref normalization>), but in general this is simply not true, as for example chapter <ref simplicity> show. And even if your working in a region where this relation is true, Classifier evaluation methods<note AUC scores even have one of the lower uncertainities> usually have a much higher uncertainity<note uncertainity in the sense that even a well trained network can change its AUC score by a couple of percent after retraining, even if it has the same loss (see chapter <ref auncertain>)> than other methods, which is why in the regions in which there is a strong correlation, it was more useful to use the loss of the network to assert that the network improves, and to simply know that the AUC score will correlate.
 
 <subsubsection title="Losses" label="evalloss">
-<ignore>So why not do this all the time: Look at the quality of the autoencoder and try to optimize only it.</ignore> Using only the quality of your autoencoder and trying to optimize this would be conceptually great, as you only need to use your anomalous data once<note usual machine learning has a problem, in which you network can learn even data that it is not trained on, simply by you comparing networks on it (this is why there is test data), the same can happen here, by you often comparing qualities of your anomalous data and since finding new test data would require you to have completely different anomalous systems, this can be difficult to do (even though we try this in chapter <ref secdata>), which is why choosing to ignore your anomalies in training would be great>, but this again has problems: Not only requires this still a strong relation between AUC and loss (That is here given even less, consider the problem of finding the best compression size: The loss will usually<note always, except for noise and random chance> fall by increasing the compression size, but at some point, the autoencoder can just reconstruct everything perfectly, and thus has no more classification potential), but the loss also relies heavily on the definition of the network and the normalization of the input data<note see chapter <ref data>>, which makes comparing different networks only possible, if you neither alter the loss nor the normalization.
+<ignore>So why not do this all the time: Look at the quality of the autoencoder and try to optimize only it.</ignore> Using only the quality of your autoencoder and trying to optimize this would be conceptually great, as you only need to use your anomalous data once<note usual machine learning has a problem, in which you network can learn even data that it is not trained on, simply by you comparing networks on it (this is why there is test data), the same can happen here, by you often comparing qualities of your anomalous data and since finding new test data would require you to have completely different anomalous systems, this can be difficult to do (even though we try this in chapter <ref secdata>), which is why choosing to ignore your anomalies in training would be great>, but this again has problems: Not only requires this still a strong relation between AUC and loss (That is here given even less, consider the problem of finding the best compression size: The loss will usually<note always, except for noise and random change> fall by increasing the compression size, but at some point, the autoencoder can just reconstruct everything perfectly, and thus has no more classification potential), but the loss also relies heavily on the definition of the network and the normalization of the input data<note see chapter <ref data>>, which makes comparing different networks only possible, if you neither alter the loss nor the normalization.
 
 <subsubsection title="Images" label="evalimg">
 This cross comparison problem can be easily solved by simply looking at the images behind the losses<note the jet image showing input and output of the autoencoder, see for an example <ref imgout>>. But while this is certainly very useful, as it also allows to understand more about your network(for example, there are networks, that simply ignore some parameters, and thus have their whole loss in those parameters, this can be most easily seen by looking at the images), this still relies on the relation between AUC and loss and more importantly is less quantitative: Giving 2 images, finding out which autoencoder is better is not always an easy task, especially since what differences you migth see in those images does not neccesarily correspond to differences the network sees<note see for this chapter <ref losses>>. Most notably you usually care more about angular differences, and mostly neglect differences in #lp_T#, while sorting by the transverse momentum introduces a sligth preference for #lp_T#.
 <ignore><i f="ptdraw632" wmode="True">some image, with nothing learned in pt</i></ignore>
 This you can solve, by also looking at the #lp_T# reproduction, but this demands weighting importance between images, and thus does not make evaluating images any easier.
 
-<subsubsection title="Oneoff width" label="evaloow">
+<subsubsection title="oneoff width" label="evaloow">
 <ignore>The probably best solution to this problem, is sadly also the least applicable, and requires that you to have read chapter (ENTER chapter) and maybe (ENTER chapter) before understanding it</ignore>
-The final solution, and the solution that seems to be the best currently, is based on the things introduced in chapter <ref secmixed>. Because of this, it will be explained in chapter <ref oneoff>. It works, by defining the loss in a way, that does not chance by chancing the loss function or the initial normalization. We do this by letting the network define its own variable, which should be constant over all background events. And by setting this constant to be #1#, the variance of this measurable is independent under chancing the inputs. This still requires some correlation between the variance of this variable and the AUC score, which we cannot assume in general, but chapter <ref impro> at least suggests that this is common.
+The final solution, and the solution that seems to be the best currently, is based on the things introduced in chapter <ref secmixed>. Because of this, it will be explained in chapter <ref oneoff>. It works, by defining the loss in a way, that does not change by changing the loss function or the initial normalization. We do this by letting the network define its own variable, which should be constant over all background events. And by setting this constant to be #1#, the variance of this measurable is independent under changing the inputs. This still requires some correlation between the variance of this variable and the AUC score, which we cannot assume in general, but chapter <ref impro> at least suggests that this is common.
 <ignore>It seems to help with all the given problems. There is a strong relation between this width and the AUC score<note or we at least did not yet find any exception to this relation, it migth well be that also this relation is also just wrong>, it is independent of the loss function and the normalization, and thus is easily comparable, while also beeing exactly what the network cares about.</ignore>
 
 
diff --git a/data/03graphae/08resultsclass b/data/03graphae/08resultsclass
index 26d550f1b1869b51fabb066d387ee8d41a34b253..821fc1e5962733c7f2732086e81f8b56ad385c2e 100644
--- a/data/03graphae/08resultsclass
+++ b/data/03graphae/08resultsclass
@@ -4,7 +4,7 @@
 <subsubsection title="4 nodes" label="class4">
 
 
-As you see in image <refi roc4>, these 4 particle networks already reach an AUC score of over #0.81#, which is quite good considering we only use 4 particles. By chancing this networks parameters you can reach AUCs upwards of #0.85#.
+As you see in image <refi roc4>, these 4 particle networks already reach an AUC score of over #0.81#, which is quite good considering we only use 4 particles. By changing this networks parameters you can reach AUCs upwards of #0.85#.
 
 <i f="recqual979" f2="roc979" wmode="True" label="roc4">Loss distribution and roc curve for the 4 particle network</i>
 
diff --git a/data/04problems/01intro/01intro b/data/04problems/01intro/01intro
index 2bc464908d65cc6709b2939f0329782b97451a2a..1bf51f41bf35837e168ba9ce14f109db0b9bbcbf 100644
--- a/data/04problems/01intro/01intro
+++ b/data/04problems/01intro/01intro
@@ -1,4 +1,4 @@
-<section title="Problems" label="secinv">
+<section title="Open questions" label="secinv">
 
 Given the results of the classifier introduced in the last chapter, there are two problems that limit the usefulness of these autoencoder. This chapter tries to understand them further, so that chapter <ref secnorm> and <ref secmixed> can solve them.
 
diff --git a/data/04problems/05inv/02simplicity b/data/04problems/05inv/02simplicity
index fda6841cb51eada2106cfd94f59b5e6269905a8a..f6a02285476e408ab30b5022d4ebeed243e9cbfc 100644
--- a/data/04problems/05inv/02simplicity
+++ b/data/04problems/05inv/02simplicity
@@ -16,9 +16,9 @@ as you see, for a low number of particles, this works fairly well, but then, at
 <i f="simponez" wmode="True" label="tscale">(tcroc/simpledraw onez) Trivial comparing angular scaling with c addition. The reason for the falloff at the end migth be the different datasetup of missing particles and the assumptions tested in chapter <ref impro></i>
 This better classifier reaches an AUC of over #0.915# that is comparable to the best anomaly detection networks, for example QCDorWhat<cite QCDorWhat> reaches #0.93#<note #0.9255# when you calculate the AUC from their roc curve> but on sligthly different data, while the work of thorben finke reached #0.908# on the same data<cite thorb>. <ignore>Not to discredit them, but you could ask yourself, if</ignore> So what is the value of those complicated models, if they only improve the AUC by at most single percentage differences<ignore>, why use a complicated model at all</ignore>.
 That beeing said, you also cannot assume that new physics has the same angular distribution difference, as QCD compared to top, making this alternative model useless in the task of finding new physics<note or at least useless unless you search for one specific kind of abnormal data, there are some examples showing other kinds of abnormal data behaving completely differently in <ref secdata>>,so the question of interrest is just: do complicated models contain something more than this trivial difference? And unfortunately this is very hard to test. C addition allows you to estimate the effect any small additional AUC would have, and an AUC of about #0.6#, optimally combined would only improve an AUC score of #0.9# to #0.904#, while #0.7# would improve up to #0.917#, so both improvements would probably be neirly unmeasurable. So there migth be some hidden effect in a hidden model, that allows them to find new physics<note please note, that we assume here a lot: first, that in #p_T# there is potential to differentiate all kinds of new physics, that this potential is used perfectly by an algorithm that did not work as expected for angles, and also, maybe most improbable, that the loss is used perfectly, and there is no confusion from the angular part at all>.
-But what we can say, is that the networks we looked at so far, probably dont do anything more than looking at angular information<note since the classifier improves when you remove momenta, and the resulting classifier again improves when you replace the returned angles by zero> and thus their fairly good AUC score is completely useless for finding new physics. What we want to do in the following is to force our network to learn something nontrivial, and thus actually create a correlation between how the network works on top jet anomalies and how it would behave on new physics.
+But what we can say, is that the networks we looked at so far, probably don`t do anything more than looking at angular information<note since the classifier improves when you remove momenta, and the resulting classifier again improves when you replace the returned angles by zero> and thus their fairly good AUC score is completely useless for finding new physics. What we want to do in the following is to force our network to learn something nontrivial, and thus actually create a correlation between how the network works on top jet anomalies and how it would behave on new physics.
 Probably the most important result of this trivial model, is the effect it has on how to evaluate a model. We already talked about why just choosing a model that has a good AUC is a bad idea in chapter <ref evalprob>, but here this could probably not be clearer:
-Since we tried to train a model to be a great seperator, when we chanced the initial normalization, we choose those, that makes the network ignore angles and focus on #p_T#. This seamed useful, since this makes the model more like the trivial one and thus get a good AUC. But this also means, that we dont have a good autoencoder anymore, since a worse reconstruction can actually improve the quality of the network.
+Since we tried to train a model to be a great seperator, when we changed the initial normalization, we choose those, that makes the network ignore angles and focus on #p_T#. This seamed useful, since this makes the model more like the trivial one and thus get a good AUC. But this also means, that we don`t have a good autoencoder anymore, since a worse reconstruction can actually improve the quality of the network.
 You can also see this in the number of nodes we use: 4 node networks seemed to result in good classifiers, and you can clearly see a pattern in the first 4 nodes in images <refi tscale> and <refi tscaleno>.
 
 
diff --git a/data/05normation/00section b/data/05normation/00section
index 797704a953fd9da56dceea5f61185a56a3045990..577c88127af8a76d67590a390cf1960c53112ca2 100644
--- a/data/05normation/00section
+++ b/data/05normation/00section
@@ -1 +1 @@
-<section title="Solution 1: normalization" label="secnorm">
+<section title="Normalization" label="secnorm">
diff --git a/data/05normation/04norm b/data/05normation/04norm
index e899364f2a001362d266aff890eedbb717dbb3d1..79e5ba85dfd239b823cfa0e88282d3034f751364 100644
--- a/data/05normation/04norm
+++ b/data/05normation/04norm
@@ -11,22 +11,22 @@ This method has one obvious drawback: Not only do we actively remove information
 
 <subsubsection title="How to normalise an autoencoder" label="normprobs">
 One thing, We did not realise before trying to normalise the input datapoints, is that simply demanding that the mean is zero and the standart deviation is one, just does not work. This migth be an effect that is most important when we talk about small networks<note networks with a low amount of input particles>, but is still somewhat of an effect in every network, and becomes important in chapters <ref oneoff> and <ref secmixed>. The problem is, that by demanding a value to be fixed, we remove the size of the input space, and by having an autoencoder that only reduces 12+flag information onto 9 values, this means, we allow the network to trivially learn 3 informations per set feature<note 3, since there are 3 variables which mean and or standart deviation we fix>, and so by setting the standart deviation and mean to be fixed, the autoencoder can trivially learn to compress 12+flag onto 6 values<note ignoring flag for now, three values is always enough to encode 4 flag values, since the four three flag values are neirly always one (the jet with the lowest number of particles has 3 particles in our trainingsset)>, which is below the size of compression space. In practice this is not so easy as there is no garantee that this minima is found, since the graph structure does not neccesarily help for this kind of transformation(see appendix <ref identities>), but training this kind of network definitely does not result in the model gaining any classification power. This can be seen in the corresponding feature map
-<i f="aucmap1011" wmode="True">(1011..maybe not the best)Aucmap for normally normated networks, showing nothing useful being learned</i>
-This migth seem now, as if there is a trivial solution: just reduce the compression size accordingly, but this has three problems
+<i f="aucmap1011" wmode="True">(1011..maybe not the best)Aucmap for normally normalized networks, showing nothing useful being learned</i>
+This seems as if there is a trivial solution: just reduce the compression size accordingly, but this has three problems
 <list>
-<e>First, it is not completely trivial to misuse the normalization (Think of the standart deviation, there is a formula giving you information about the 4th value, given the first three. But even if we ignore the mean as beeing 0, this formula still involves squares and roots, which the network has to learn, and even then, there are always two possibilities for the resulting value.). So assuming that this is trivial, and that the network will always learn it garantueed, would be wrong</e>
-<e>Even if this is learned, this would not be enough: the network still has to compress this information further and this can lead to situations in which the network has to decide between learning the easy compression and the learning the interresting compression. In this situations it will probably always learn the trivial one</e>
-<e>Trying to compress data with removed information further is not as easy as compressing non removed information. Think of two values distributed between #0# and #10#: For example #4# and #6#, or, after setting the mean to be #0#: #-1# and #1#. In #4# and #6# the network can still decide to just average both values and get an mediocre prediction of #5#, which still describes these values in a way. But if after removing the mean, the network still averages both values, the predicted #0# is fairly useless<note please note, that the difference between a good normalization and a bad one is just physical intuition. For example, we still set the mean of the jet angles to be zero, just because the direction relative to the measurement should be unimportant>. From this we conclude, that simply subtracting each fixed value from the compression size does not work, as we would expect less good classifier.</e>
+<e>First, it is not completely trivial to misuse the normalization (Think of the standart deviation, there is a formula giving you information about the 4th value, given the first three. But even if we ignore the mean as beeing 0, this formula still involves squares and roots, which the network has to learn, and even then, there are always two possibilities for the resulting value.). So assuming that this is trivial, and that the network will always learn it garantueed, would be wrong.</e>
+<e>Even if this is learned, this would not be enough: the network still has to compress this information further and this can lead to situations in which the network has to decide between learning the easy compression and the learning the interresting compression. In this situations it will probably always learn the trivial one.</e>
+<e>Trying to compress data with removed information further is not as easy as compressing non removed information. Think of two values distributed between #0# and #10#: For example #4# and #6#, or, after setting the mean to be #0#: #-1# and #1#. In #4# and #6# the network can still decide to just average both values and get an mediocre prediction of #5#, which still describes these values in a way. But if after removing the mean, the network still averages both values, the predicted #0# is fairly unproductive<note note, that the difference between a good normalization and a bad one is just physical intuition. For example, we still set the mean of the jet angles to be zero, just because the direction relative to the measurement should be unimportant>. From this we conclude, that simply subtracting each fixed value from the compression size does not work, as we would expect less good classifier.</e>
 </list>
 
 So what we need, is a better way of normalising the input data. From our thougths above, this new method should satisfy these conditions:
 <list>
 <e>Translation invariance: #Eq(n(x),n(x+a))#</e>
 <e>Scale invariance:#Eq(n(x),n(a*x))#</e>
-<e>No fixed features: you can not write any #f(x)# so that #Eq(f(n(x)),0)# for every #x# is given</e>
+<e>No fixed features: you can not write any #f(x)# so that #Eq(f(n(x)),0)# for every #x# is given.</e>
 </list>
 
-The first two rules are obvious, since we want to use this, to remove any size information, and third rule would solve the problem of an autoencoder focussing on normalization artefacts<note this last rule would actually be solved by demanding the standart deviation to be constant>.
+The first two rules are obvious, since we want to use this, to remove any size information, and the third rule would solve the problem of an autoencoder focussing on normalization artefacts<note this last rule would actually be solved by demanding the standart deviation to be constant>.
 
 All three rules<note except for scale invariance with #LessThan(a,0)#> are solved by the following 3 normalization steps (#x# is the input, #n# the output of the normalization method)
 
@@ -36,18 +36,18 @@ All three rules<note except for scale invariance with #LessThan(a,0)#> are solve
 
 Here the definitions of #y# and #n# assert translation and scale invariance respectively, while #n# and #z# remove any kind of artefacts. Why they do this can be easily understand for #n#, by diving through the maximum value instead of the standart deviation: The only relation given is, that if none of the first three values is either #+1# or #-1#, the last value is either #+1# or #-1#. And even if we ignore that misusing this relation would be quite complicated to implement for a neural network <note there are uncertainities: if you have a value of #0.997#, is there still a #1# remaining?> there is no way to differenciate between #1# and #-1# in general<note you could say, that this normalization uses the problem of mean reproducing networks (chapter <ref ameans>) to its benefit, by making the errors of a #1# or #-1# guessing network bigger than every other possible distance. But this is a bit more complicated through the definition of #z#, since this kind of outputs have a clearly negative bias>. Also dividing through the maximum value is generally a good idea compared to dividing through the standart deviation, since it only divides by zero when every value is zero, and not when every value is the same, which is more probable<note there is a small constant in our definition to remove this divergences, but this still removes some big gradients>. Also dividing by the maximum is generally a bit faster, while resulting in less nans (chapter <ref nans>). The other definition is less easily understandable: First note, that it does not violate the first two rules, since #y# is already translation invariant, #z# is too, and since every #z(y)# is scale invariant, Scale invariance is also given<note you migth notice, that this is only the case for positive multiplicators, but we think this is an acceptable compromise for fullfilling rule 3. Also you could also argue, that we only want to remove nonphysical information, and since for #p_T# this is not a problem (since #p_T# is positive and it has a peculiar shape), this is only interresting for angular information, and parity is broken>. Now consider only the definition of #z# and two different vectors #y# given by #Matrix([[1,0]])# and #Matrix([[1,1]])#. Both result in the same first component, but different second components, so there is no information from the first value to the second, and thus rule 3 is satisfied. Less theoretically, a network using this kind of normalization has classification power:
 
-<ignore><i f="aucmap928" wmode="True">(928..definitely take some top one)An AUC map for a better normated network</i></ignore>
+<ignore><i f="aucmap928" wmode="True">(928..definitely take some top one)An AUC map for a better normalized network</i></ignore>
 
 <subsubsection title="Using this normalization" label="usenorm">
 Using this kind of normalization, 4 node networks are invertible. And not only this, but also most features are invertible.
 <i f="aucmap677" f2="aucmap928" wmode="True">invertible 4node network auc maps achieved by a better normalization</i>
 But the quality suffers
-<i f="drtoptagging" wmode="True">(generate later for computational reasons)double roc curve for invertibility of normated networks</i>
+<i f="drtoptagging" wmode="True">(generate later for computational reasons)double roc curve for invertibility of normalized networks</i>
 and there are other consequences of the fact that this network actually has to learn something nontrivial:
 First, we were forced to increase the size of the compressed feature space from 5 to 9. This makes sense, as a network that compares angles to zero, has to just reconstruct zeros in each angle, and thus only has to save #p_T#<note and maybe flag, but as seen later in this chapter and more in chapter <ref oneoff>, this is usually not actually be the case>, needing only a smaller latent space.
 Also networks, that before were very reproducable in their training<note which makes sense, as they always just needed to learn to ignored the angles> are now less stable, and often vary their loss<note remember that the l2 loss goes quadratic in changes of the inputs> over about one order of magnitude. Interrestingly, this variation shows a clear relation between the loss, and the classification quality
 <i f="rtvl 0 2 4 5 6 7 8 9" wmode="True">(from c3pp atm)Relation between the network loss and the AUC score</i>
-This relation is great, since it means, that finding a better autoencoder, automatically results in a better classifier, and we thus can focus completely on improving the autoencoder. 
+This relation is very useful, since it means, that finding a better autoencoder, automatically results in a better classifier, and we thus can focus completely on improving the autoencoder. 
 Also by looking at this relation, we are able to justify the new compression size
 <i f="none" wmode="True">(needs to be generated a bit later sadly for computational reasons)justification of the new compression size, auc vs loss for compression size 8 and 9</i>
 since this is the first compression size, at which the network becomes invertible.
diff --git a/data/05normation/06aucs b/data/05normation/06aucs
index 55dc18f650d45fa9c8985754c76b8270aa9f2751..3e8de1ba072cbbcb359ed58c03a9211aad41b25e 100644
--- a/data/05normation/06aucs
+++ b/data/05normation/06aucs
@@ -1,12 +1,12 @@
-<subsubsection title="Improving the AUC scores for normated networks" label="normimpro">
+<subsubsection title="Improving the AUC scores for normalized networks" label="normimpro">
 
-These initial normated networks are not very good. This migth be what we expected, since we remove trivial information, but we still are able to improveon them quite a bit. Namely by using the exact model setups and training parameters from <ref setup> with one additional normalization layer before the first comparison value<note you migth be quite a bit confused, why we chose other models but those that work well for unnormated networks to test our normalization, but this is just a problem of way to many just sligthly varying network setups: We used more quite different unnormated networks, but since learning zeros does not depend to much on network parameters, we simply use the new normated network setups for both networks, to not need to explain both>.
+These initial normalized networks are not very good. This migth be what we expected, since we remove trivial information, but we still are able to improveon them quite a bit. Namely by using the exact model setups and training parameters from <ref setup> with one additional normalization layer before the first comparison value<note you migth be quite a bit confused, why we chose other models but those that work well for unnormalized networks to test our normalization, but this is just a problem of way to many just sligthly varying network setups: We used more quite different unnormalized networks, but since learning zeros does not depend to much on network parameters, we simply use the new normalized network setups for both networks, to not need to explain both>.
 Using this we are able to improve the network trained on top up to #0.377#.
 
-<subsubsection title="Scaling in normated networks" label="scalenorm">
-Sadly this normalization does not chance scaling problems to much. Bigger networks still contain more trivial information, since the number of parameters fixed is constant, and even when using batches to scale, the invertibility is just a feature of the first batch
+<subsubsection title="Scaling in normalized networks" label="scalenorm">
+Sadly this normalization does not change scaling problems to much. Bigger networks still contain more trivial information, since the number of parameters fixed is constant, and even when using batches to scale, the invertibility is just a feature of the first batch
 
-<i f="m4scaleroc" wmode="True">(multi4scale roc) AUC values for higher normated batches by their training data</i>
+<i f="m4scaleroc" wmode="True">(multi4scale roc) AUC values for higher normalized batches by their training data</i>
 
 
 <ignore>
@@ -14,7 +14,7 @@ Sadly this normalization does not chance scaling problems to much. Bigger networ
 
 HAVE I ALREADE WRITTEN THIS?
 
-These normated networks do result in way worse AUC scores, as expected by removing the probably most interresting feature. Initial tries result in AUC scores around #0.55#, but interrestingly they can be invertible. This is not as easy as just switching background for signal data, as the old low compression size of #5# of #12+flag# does not allow for any useful reconstruction (as would have been expected, since the networks that dont reconstruct angles, obviously require less compression size), but from a compression size of #9# of #12+flag# Features seem to be the first compression size that allows for invertible networks
+These normalized networks do result in way worse AUC scores, as expected by removing the probably most interresting feature. Initial tries result in AUC scores around #0.55#, but interrestingly they can be invertible. This is not as easy as just switching background for signal data, as the old low compression size of #5# of #12+flag# does not allow for any useful reconstruction (as would have been expected, since the networks that don`t reconstruct angles, obviously require less compression size), but from a compression size of #9# of #12+flag# Features seem to be the first compression size that allows for invertible networks
 
 <i f="totalcomp0" wmode="True">(c3p/totalroccomp.py)Invertibility as function of compression, showing also in accuracy</i>
 
diff --git a/data/05normation/08scaling b/data/05normation/08scaling
index af5ecab0f9f610e362bc94bbbfbc3b045d680a64..9bc8906c1fdff69f3074918d9fd6c962100ffa7c 100644
--- a/data/05normation/08scaling
+++ b/data/05normation/08scaling
@@ -1,5 +1,5 @@
 <ignore>
-<subsection title="Scaling in normated networks" label="scale2">
+<subsection title="Scaling in normalized networks" label="scale2">
 
 (STILL MISSING)
 
diff --git a/data/05normation/09normplus b/data/05normation/09normplus
index 130ae6ffdbe255ef65b15614604eea9311f92b74..a4e6d0ea7100a8640381eb7c26e2d89a9dd51e10 100644
--- a/data/05normation/09normplus
+++ b/data/05normation/09normplus
@@ -1,14 +1,14 @@
 
 <subsubsection title="Improving the normalization even further" label="normplus">
-After seeing what an effect some kind of normalization can have, we are not completely satisfied anymore with the normated feature maps:
+After seeing what an effect some kind of normalization can have, we are not completely satisfied anymore with the normalized feature maps:
 <i f="aucmap928" wmode="True" wid="0.8">ABE for better norm (just a copy)</i>
 consider the highest #p_T# Value (the lower rigth corner). While beeing the generally most interresting particle, there is no classification power in it, and by looking at its distribution it becomes clear why
 <i f="pt0draw928" wmode="True" wid="0.8">(928/drawp0.py)distribution of the transverse momentum of the first particle</i>
-These values are basically constant, so its input it the same as the flag values (first collumn), from which we dont expect any physically useful information.
+These values are basically constant, so its input it the same as the flag values (first collumn), from which we don`t expect any physically useful information.
 So lets solve this: Since #lp_T# mostly has the same structure<note to be more precise, the difference between the first and the second particle is higher than the difference between the last two ones>, most jets transverse momentum get divided by the first one, resulting in it always having the same value. We solve this by replacing the definition of #n# in chapter <ref normalization> to be:
 ##Eq(n,2*z/(max(abs(z))+mean(abs(z))))##
 
 removing the need to set one value to either positive or negative one, and thus making the highest value in #lp_T# actually useful, and as you see, this removes the difference in the #lp_T# AUC.
-<i f="aucmap534" wmode="True" wid="0.8">ABE good norm</i>
+<i f="aucmap534" wmode="True" wid="0.8">AUC feature map for a well normated network</i>
 But, as you also see, now the whole classification power lies now in flag, and this should be quite confusing to you: Something having no physical meaning beeing more useful than everything else. Not to different compared to chapter <ref simplicity>.
 This we will explain in chapter <ref oneoff>.
diff --git a/data/06othernets/00intro b/data/06othernets/00intro
index 4a250141414820d31163baef112a7e34d63cc90b..c76f87105875fa2a4eb1a407e5e846f04aeed37d 100644
--- a/data/06othernets/00intro
+++ b/data/06othernets/00intro
@@ -1,5 +1,5 @@
-<section title="Solution 2: Mixed networks" label="secmixed">
+<section title="Mixed networks" label="secmixed">
 
 <ignore>
-Having used the last chapter to intoduce oneOff networks, while not beeing able to use them for jet tagging, this chapter finds usage for them, improving autoencoder by a clear margin.
+Having used the last chapter to intoduce oneoff networks, while not beeing able to use them for jet tagging, this chapter finds usage for them, improving autoencoder by a clear margin.
 </ignore>
\ No newline at end of file
diff --git a/data/06othernets/02oneoff b/data/06othernets/02oneoff
index 4ae35f6c7840b05162d8b82feae5f111fc38ad03..eb50d4bed33746b40a732d90120cf7797fc89871 100644
--- a/data/06othernets/02oneoff
+++ b/data/06othernets/02oneoff
@@ -1,26 +1,26 @@
-<subsection title="Oneoff networks" label="oneoff">
+<subsection title="oneoff networks" label="oneoff">
 
-When you consider the following feature map of a well normated network:
+When you consider the following feature map of a well normalized network:
 <i f="aucmapb" wmode="True" wid="0.8">(534)AUC Feature map for an on top trained autoencoder, using a good normalization</i>
-You see, that most of the decision power is in the first feature, but the first feature, flag is basically just 1<note>flag is 1 as long as the current event does not contain less particles than the network demands, and since this is a network with only 4 nodes, and there are very few jets with only 3 particles or even less, saying flag is a constant (#Eq(flag,1)#) is a quite good approximation</note>. This migth seem a bit counterintuitive or unphysical at first, how can a variable without any physical meaning be a better seperator than those variables with physical meaning?
-To explain this, we need to take a bit more close look at what the network is doing: First, just because the output is has no physical meaning, this does not mean, that no physical variables are used in its calculcation. In fact, before this, we always just assumed that there is one parameter in the latent space, that is learned to be just a one from the input space<note>This is a bit of a simplification, most importantly it would be untestable, since instead of learning a constant, the network could learn a constant as a function of multiple parameters (for a simple example consider #Eq(x_2,x_1+1)#, both variables are not constant, making it harder to find this, but still #x_2# has no additional information with respect to #x_1#, and there is a one learned as #x_2-x_1#)</note>, but this distributiuon of decisionpower implies that this is not the case: If there would be a constant feature in the compression space, the constant output would be a trivial copy of this constant and thus have no physical meaning. More likely is the following: The network is able to reconstruct an #1# from all the other parameters. This makes sence, since we got this AUC distribution by chancing the normalization in a way that made trivial ones in the input space much less likely<note>Since we stopped dividing by #max(abs(x))# and started dividing by #(max(abs(x))+mean(abs(x)))/2#, it is no longer the case that there is either a #-1# or a #1# in each feature</note>, and it also explains how an unphysical output can be physically useful: Since the are utilizing physical inputs, the resulting constant has to be a function of the inputs. And when you chance the inputs, the constant is also chanced and this chance we can use to differenciate signal and background events.
-And since this quality is better than every other autoencoder decision quality, it migth be useful to use this: If appearently nonphysical outputs can be at least as good as physical outputs, why not just use outputs that are nonphysical (Outputs that are one). This is what we call oneOff networks<note>Since the distance off 1 is the deciding quality indicator and it is a oneClass algorithm</note>, and on paper it seams like a great idea: As shown before (see chapter <ref simplicity>), complexity is to a big part just width. You may be able to solve this by normalization, but this removes information, and oneOff networks would not require this<note>Since their output, #1# , is obviously automatically normated</note><note>Also in practice it seems to be still a good idea to normate also oneoff networks, this migth be because this normalization also lets features the oneoff network focusses on to be more similar and thus easier to combine, or because similar sized inputs are easier to train on</note>. Also there migth be a certain kind of complexity benefit, since the whole network is made to just minimize one distance<note> Actually, in practice it seems to simplify the training, if you dont use only one output, but multiple ones, that all are compared to 1 and which mean is used. This results in very high correlations in the outputs, but seems to help in the convergence of the network</note> that is always the same, instead of optimizing some feature that migth be useful for some events, but weakenen it while considering other events, in which this feature plays a less important role. This should result in the network beeing able to learn more complicated functions.
+You see, that most of the decision power is in the first feature, but the first feature, flag is basically just one<note>flag is 1 as long as the current event does not contain less particles than the network demands, and since this is a network with only 4 nodes, and there are very few jets with only 3 particles or even less, saying flag is a constant (#Eq(flag,1)#) is a quite good approximation</note>. This migth seem a bit counterintuitive or unphysical at first, how can a variable without any physical meaning be a better seperator than those variables with physical meaning.
+To explain this, we need to take a bit more close look at what the network is doing: First, just because the output is has no physical meaning, this does not mean, that no physical variables are used in its calculcation. In fact, before this, we always just assumed that there is one parameter in the latent space, that is learned to be just a one from the input space<note>This is a bit of a simplification, most importantly it would be untestable, since instead of learning a constant, the network could learn a constant as a function of multiple parameters (for a simple example consider #Eq(x_2,x_1+1)#, both variables are not constant, making it harder to find this, but still #x_2# has no additional information with respect to #x_1#, and there is a one learned as #x_2-x_1#)</note>, but this distributiuon of decision power implies that this is not the case: If there would be a constant feature in the compression space, the constant output would be a trivial copy of this constant and thus have no physical meaning. More likely is the following: The network is able to reconstruct an #1# from all the other parameters. This makes sence, since we got this AUC distribution by changing the normalization in a way that made trivial ones in the input space much less likely<note>Since we stopped dividing by #max(abs(x))# and started dividing by #(max(abs(x))+mean(abs(x)))/2#, it is no longer the case that there is either a #-1# or a #1# in each feature</note>, and it also explains how an unphysical output can be physically useful: Since the are utilizing physical inputs, the resulting constant has to be a function of the inputs. And when you change the inputs, the constant is also changed and this change we can use to differenciate signal and background events.
+And since this quality is better than every other autoencoder decision quality, it migth be useful to use this: If appearently nonphysical outputs can be at least as good as physical outputs, why not just use outputs that are nonphysical (Outputs that are one). This is what we call oneoff networks<note>Since the distance off 1 is the deciding quality indicator and it is a oneClass algorithm</note><ignore>, and on paper it seams like a great idea</ignore>: As shown before (see chapter <ref simplicity>), complexity is to a big part just width. You may be able to solve this by normalization, but this removes information, and oneoff networks would not require this<note>Since their output, #1# , is obviously automatically normalized</note><note>Also in practice it seems to be still a good idea to normate also oneoff networks, this migth be because this normalization also lets features the oneoff network focusses on to be more similar and thus easier to combine, or because similar sized inputs are easier to train on</note>. Also there migth be a certain kind of complexity benefit, since the whole network is made to just minimize one distance<note> Actually, in practice it seems to simplify the training, if you don`t use only one output, but multiple ones, that all are compared to 1 and which mean is used. This results in very high correlations in the outputs, but seems to help in the convergence of the network</note> that is always the same, instead of optimizing some feature that migth be useful for some events, but weakenen it while considering other events, in which this feature plays a less important role. This should result in the network beeing able to learn more complicated functions.
 
 We justify this idea mathematically in appendix <ref oomath> and <ref impro>
 
 
 <subsubsection title="oneoff quality" label="ooquality">
 
-So lets try this out: A simple dense network with just an output that should be one, sadly still has a lot of problems.
-First: the loss can go to basically zero(#10**(-12)#), which is a bit unphysical, since the loss, as a distance to one, is basically the variance of the used feature, and you would not expect there to be any physically significant feature of this accuracy in 4 particles<note>Especially, since the lowest difference there can be in the used float32 implementation is bigger than #10**(-8)# and thus, since the final loss is the mean of each loss, this would mean, that at least #0.9999# of each event reproduce exactly 1</note>. So there are features that are more trivial to learn, and make any decision process meaningless. And it is not neccesarily trivial to find those, there migth be those features that are just input variables of one (for example an input that would be set to flag), but not all of them are that easy to find. <note>A notable example migth be the preprocessing of #lp_T#. As descibed in chapter <ref data>, we used a preprocessing similar to that of particleNet: #Eq(x,ln(p_Tjet/p_T))#, but this means (because of the implementation), that a sum over #exp(-x)# is always #1#. This migth be a good time to talk about functions in those kind of networks. Since we have to forbidden any biases (a bias would just result in the network learning a zero and adding a one as bias), the usual reason for a network to learn any function has to be modified a bit. Think about taylor approximations: A function like #exp(x)# could be written as #1+x+O(x**2)# (with as many term as the networks needs), but for a network to learn #1#, the input of #exp(x)# would then be learned to zero, the network would be one and it is basically the same as adding a constant bias. But adding a bias is not allowed, and thus the network can not learn #exp(x)#, but the network can learn #Eq(exp(x)-1,x+O(x**2))#, and, when #Eq(sum(exp(-x_i),i),1)# then is #Eq(sum(exp(-x_i)-1),-3)# for 4 nodes, and thus the network can learn this, without having learned anything physically useful</note>. This means, that training an oneoff network is a bit like outsmarting your algorithm. One thing that we found quite useful, is letting the network not only learn a one on the data that you are interrested in, but also zero on other random data.<note>We choose here random events with the same mean and standart deviation in each feature, as the original data, that still goes through the same preprocessing</note>. When we use relu activations here<note>Activations are another thing where those networks can become trivial, think of a sigmoid and a network just learning infinite values before activation</note>, learning values to be zero, means learning them just to be negative, and is thus way easier. This can demand that the network does not fixate on trivial features in the networksetup and preprocessing<note> later on, in chapter <ref mixedidea>, this is no longer needed, and just complicates the training</note>.
-A simple oneoff network reaches usually an auc of at best #0.6# for the task of finding top jets, which is not to impressive. But if you look at the classification power as a function of the training epoch, you see that this only is so bad, since those AUC scores is way better at earlier epochs
-<i f="mabe3" wmode="True" wid="0.8">(multiabe 3)Auc as function of the epoch, trained on QCD, once for a graph oneOff and once for a dense oneOff. As you see, both relations show a maximum before the training ends, but the graph network is way more continuos</i>
+A simple dense network with just an output that should be one, sadly still has a lot of problems.
+First: the loss can go to basically zero(#10**(-12)#), which is a bit unphysical, since the loss, as a distance to one, is basically the variance of the used feature, and you would not expect there to be any physically significant feature of this accuracy in 4 particles<note>Especially, since the lowest difference there can be in the used float32 implementation is bigger than #10**(-8)# and thus, since the final loss is the mean of each loss, this would mean, that at least #0.9999# of each event reproduce exactly 1</note>. So there are features that are more trivial to learn, and make any decision process meaningless. And it is not neccesarily trivial to find those, there migth be those features that are just input variables of one (for example an input that would be set to flag), but not all of them are that easy to find. <note>A notable example migth be the preprocessing of #lp_T#. As descibed in chapter <ref data>, we used a preprocessing similar to that of particleNet: #Eq(x,ln(p_Tjet/p_T))#, but this means (because of the implementation), that a sum over #exp(-x)# is always #1#. This migth be a good time to talk about functions in those kind of networks. Since we have to forbidden any biases (a bias would just result in the network learning a zero and adding a one as bias), the usual reason for a network to learn any function has to be modified a bit. Think about taylor approximations: A function like #exp(x)# could be written as #1+x+O(x**2)# (with as many term as the networks needs), but for a network to learn #1#, the input of #exp(x)# would then be learned to zero, the network would be one and it is basically the same as adding a constant bias. But adding a bias is not allowed, and thus the network can not learn #exp(x)#, but the network can learn #Eq(exp(x)-1,x+O(x**2))#, and, when #Eq(sum(exp(-x_i),i),1)# then is #Eq(sum(exp(-x_i)-1),-3)# for 4 nodes, and thus the network can learn this, without having learned anything physically useful</note>. This means, that training an oneoff network is a bit like outsmarting your algorithm. One thing that we found quite useful, is letting the network not only learn a one on the data that you are interrested in, but also zero on other random data.<note>We choose here random events with the same mean and standart deviation in each feature, as the original data, that still goes through the same preprocessing</note>. When we use relu<note A relu activation can be defined as #x+abs(x)#. See Appendix <ref arelu> for why this is useful> activations here<note>Activations are another thing where those networks can become trivial, think of a sigmoid and a network just learning infinite values before activation</note>, learning values to be zero, means learning them just to be negative, and is thus way easier. This can demand that the network does not fixate on trivial features in the networksetup and preprocessing<note> later on, in chapter <ref mixedidea>, this is no longer needed, and just complicates the training</note>.
+A simple oneoff network reaches usually an auc of at best #0.6# for the task of finding top jets, which is not to impressive. But if you look at the classification power as a function of the training epoch, you see that this only is so bad, since those AUC scores is way better at earlier epochs.
+<i f="mabe3" wmode="True" wid="0.8">(multiabe 3)Auc as function of the epoch, trained on QCD, once for a graph oneoff and once for a dense oneoff. As you see, both relations show a maximum before the training ends, but the graph network is way more continuous</i>
 Sadly, this observation is not really useful, since stopping the training at the optimal epoch would not be unsupervised. It is still quite interresting, since it shows, that there is some potential in those kind of networks, which is just not utilised good enough<note>this will be solved in chapter <ref mixedidea></note>.
 Another problem is again invertibility: It is possible to create an invertible oneoff network, but it is not trivially given. This becomes easier, when you use a lot of parameters. To do this, a graph network is less useful, than just a simple dense network.
 
 
-Even though they are not yet appliable here, we show in appendix <ref oomnist> that oneOff networks are very useful for finding anomalies in other datasets. This allows us to suggests that combining multiple oneOff retrains can increase the classification power even further.
-We also show that you can use oneOff networks to extract human readable information from physical events in appendix <ref oometrik>.
+Even though they are not yet appliable here, we show in appendix <ref oomnist> that oneoff networks are very useful for finding anomalies in other datasets. This allows us to suggests that combining multiple oneoff retrains can increase the classification power even further.
+We also show that you can use oneoff networks to extract human readable information from physical events in appendix <ref oometrik>.
 
 
 
diff --git a/data/06othernets/03idea b/data/06othernets/03idea
index a8aaa5386e96d12d7bf01be04d37a531b1a0b070..f84ed6e4b68f0d88a8a70b9a06b899a02f54b917 100644
--- a/data/06othernets/03idea
+++ b/data/06othernets/03idea
@@ -1,7 +1,7 @@
-<subsection title="Compressed oneOff learning" label="mixedidea">
+<subsection title="Compressed oneoff learning" label="mixedidea">
 
-The main problem of autoencoder migth be the fact that its loss function is not neccesarily the best possible seperator<note see chapter <ref losses> and chapter <ref secinv>>, while the problem of oneOff networks seem to be that they focus on useless information, which keeps them from reaching their optimum<note see chapter <ref oneoff>><ignore>other methods seems to be the fact that they are not well equiped to handle symmetries of certain kind or trivial information<note in the case of oneoffs></ignore>, but maybe combining both methods could solve both problems: You train an autoencoder to convert the input space into the latent space, to run an oneOff algorithm on this compressed space<note we also tried alternative algorithms, but oneOff networks result in the best results, see for this appendix <ref mixedalt> and <ref other>>. This means that the seperatorion function is now quite good, and the autoencoder can filter out trivial inputs. The idea of combining networks is not exactly new<note see for example <cite latentspace>>. It also is harder to train, since we now have two independent networks: Something that improves the first network migth hurt the second, but in practice this works quite well.
-It is not yet clear if you want to train your autoencoder on the background data or on both the signal and the background. Here we train on background data, since every bit of higher inaccuracy that migth be reached by giving the original autoencoder unknown data, will help the following algorithm, but the effect of chancing this is tiny anyway. Also working as unsupervised as possible is not so easy: Defining a set with absolutely no anomalies is not completely unsupervised, but defining a set that is exactly half abnormal migth be worse: The anomalies we search are probably quite rare, and approximating this fraction as #0# seems to be more realistic than approximating it as #0.5#.
+The main problem of autoencoder migth be the fact that its loss function is not neccesarily the best possible seperator<note see chapter <ref losses> and chapter <ref secinv>>, while the problem of oneoff networks seem to be that they focus on useless information, which keeps them from reaching their optimum<note see chapter <ref oneoff>><ignore>other methods seems to be the fact that they are not well equiped to handle symmetries of certain kind or trivial information<note in the case of oneoffs></ignore>, but maybe combining both methods could solve both problems: You train an autoencoder to convert the input space into the latent space, to run an oneoff algorithm on this compressed space<note we also tried alternative algorithms, but oneoff networks result in the best results, see for this appendix <ref mixedalt> and <ref other>>. This means that the seperatorion function is now quite good, and the autoencoder can filter out trivial inputs. The idea of combining networks is not exactly new<note see for example <cite latentspace>>. It also is harder to train, since we now have two independent networks: Something that improves the first network migth hurt the second, but in practice this works quite well.
+It is not yet clear if you want to train your autoencoder on the background data or on both the signal and the background. Here we train on background data, since every bit of higher inaccuracy that migth be reached by giving the original autoencoder unknown data, will help the following algorithm, but the effect of changing this is tiny anyway. Also working as unsupervised as possible is not so easy: Defining a set with absolutely no anomalies is not completely unsupervised, but defining a set that is exactly half abnormal migth be worse: The anomalies we search are probably quite rare, and approximating this fraction as #0# seems to be more realistic than approximating it as #0.5#.
 
 
 
diff --git a/data/06othernets/04areallygood b/data/06othernets/04areallygood
index e27ff30616d9780f9fc690c486acec1e3eff8903..bd84cdb317f8715890484c3217d6121c287e1e1c 100644
--- a/data/06othernets/04areallygood
+++ b/data/06othernets/04areallygood
@@ -1,21 +1,21 @@
 <subsection title="A good classifier" label="finalae">
 
 With the same setup as before (see chapter <ref setup>) and normation as well as after training 25 oneoff networks on each latent space we gain the final top tagger for this thesis
-<subsubsection title="trained on QCD" label="classQCD">
+<subsubsection title="Trained on QCD" label="classQCD">
 
-<i f="sephist928" f2="seproc928" wmode="True">oneoff loss distribution and roc curve for a network trained on top jets</i>
+<i f="sephist928" f2="seproc928" wmode="True">Oneoff loss distribution and roc curve for a network trained on top jets</i>
 
 We still see AUCs worse than in chapter <ref secgae>, but consistently better than by using just normation
 
 Interrestingly this also helps the reconstruction quality
-<i f="simpledraw928" f2="ptdraw928" wmode="True">Reconstruction images for a normated network trained on QCD</i>
+<i f="simpledraw928" f2="ptdraw928" wmode="True">Reconstruction images for a normalized network trained on QCD</i>
 
 
-<subsubsection title="trained on top" label="classtop">
+<subsubsection title="Trained on top" label="classtop">
 
 Trained on top this improves quite a lot.
 <i f="sephist1128" f2="seproc1128" wmode="True">oneoff loss distribution and roc curve for a network trained on top jets</i>
 
 Also here the reconstruction quality improves
 
-<i f="simpledraw1128" f2="ptdraw1128" wmode="True">Reconstruction images for a normated network trained on top</i>
+<i f="simpledraw1128" f2="ptdraw1128" wmode="True">Reconstruction images for a normalized network trained on top</i>
diff --git a/data/06othernets/06scale b/data/06othernets/06scale
index 1d5fc3a34713c9bd770d82b433e9b750be9d16f7..13244ad92fa196f0efb2a5b5e4550d22faf372b3 100644
--- a/data/06othernets/06scale
+++ b/data/06othernets/06scale
@@ -1,8 +1,8 @@
-<subsection title="Scale" label="scale3">
+<subsection title="Scaling for oneoff networks" label="scale3">
 
-OneOffs still dont solve the problem of different parts of the network beeing added supoptimally. You see this when you consider this 9 node network trained on top jets:
+oneoffs still don`t solve the problem of different parts of the network beeing added supoptimally. You see this when you consider this 9 node network trained on top jets:
 <i f="simpledraw1404" f2="ptdraw1404" wmode="True"> Reconstruction image for a 9 node network on top</i>
 Even though its reconstruction is much better than those from chapter <ref secgae>, we also see here that its AUC falls compared to the 4 node alternative: It reaches only and AUC of #0.34# compared to the #0.177# from the 4 node one.
 On the other hand, the batches considered in chapter <ref normimpro> are now all invertible
 <i f="m4scalesep" wmode="True"> Invertibility of batches in oneoff networks</i>
-Here you see a much more interresting relation compared to before. The variance grows with the batch index, which is expected, but some networks actually beat the AUC score of the first batch (batch 3 has a event below #0.15#). This is a result of the number of particles in each jet becoming a feature at some point. You see this, by noticing that the relation between auc and batch number is not linear: The AUCs for the second batch migth even be some of the worst, even though they should have the second most information next the the first batch.
\ No newline at end of file
+Here you see a much more interresting relation compared to before. The variance grows with the batch index, which is expected, but some networks actually beat the AUC score of the first batch (batch 3 has a event below #0.15#). This is a result of the number of particles in each jet becoming a feature at some point. You see this, by noticing that the relation between AUC and batch number is not linear: The AUCs for the second batch migth even be some of the worst, even though they should have the second most information next the the first batch.
diff --git a/data/07otherdata/01intro b/data/07otherdata/01intro
index 414b2ce10262a7f30cf48b21b36f1adcf3d739d0..6a69664a04f71a04e0b33877d412e37002402d04 100644
--- a/data/07otherdata/01intro
+++ b/data/07otherdata/01intro
@@ -1,11 +1,11 @@
-<section title="Other data" label="secdata">
+<section title="Applying this model to other datasets" label="secdata">
 
 We migth be able to supervisedly seperate probably any kind of data, but as shown in the previous chapters, if we remove the labels, this exercise becomes a lot harder. Still, usually this problem is stated as follows: Given a set of datapoints, can we write an algorithm to detect a second set of datapoints. The only difference to the supervised case is the fact that we cannot look at the anomaly set in training. And information from the validation dataset can leak into the model setup<note see <cite leakage>>.<ignore> In the training of your algorithm, you migth not look at the signal dataset, but after training each model, your decision certainly is influenced by the results on this dataset and so your anomalies leak into your model until in the worst case, you migth be able to find these anomalies, but not much else. </ignore>
-This is an effect, that is usually solved by introducing test data. Data that is only used once at the end of your analysis: If your network works worse on this data, then your setup contains information about your validation data and is no longer as general. We want to use this chapter to introduce our test data. The difference here is, that we cannot simply use some part of our validation set: It are not the events of the validation set that are leaking<note as shown in <ref evalae> our networks dont overfit>, but the specifics of your anomalies. So as test set, we have to use completely different anomalies<note training on your anomalies to find your background can help, but even this can not really exclude that your data leaks into your model( if your signal and background differ completely in one parameter, optimizing both ways would result in a network only focussing on this parameter, and thus not beeing very general), so the only real way migth be to train on as much datasets as possible, and demand that all work.>.
+This is an effect, that is usually solved by introducing test data. Data that is only used once at the end of your analysis: If your network works worse on this data, then your setup contains information about your validation data and is no longer as general. We want to use this chapter to introduce our test data. The difference here is, that we cannot simply use some part of our validation set: It are not the events of the validation set that are leaking<note as shown in <ref evalae> our networks don`t overfit>, but the specifics of your anomalies. So as test set, we have to use completely different anomalies<note training on your anomalies to find your background can help, but even this can not really exclude that your data leaks into your model( if your signal and background differ completely in one parameter, optimizing both ways would result in a network only focussing on this parameter, and thus not beeing very general), so the only real way migth be to train on as much datasets as possible, and demand that all work.>.
 <ignore>
 So we cannot really be sure, that our algorithm is general enough to work on other anomalies</ignore>
 <ignore>Sure, taking alternative approaches as suggested in chapter <ref secother> migth help, but at the end, you have to solve your actual problem, of not finding some signal data, but every other dataset that is not equal to your original one.</ignore>
-You could see this, as chancing our initial task: Instead of finding one specific anomaly, we now want to find every other anomaly.
-Please take a moment to notice the huge difference in complexity of this task: Defining every alternative dataset as signals is not solved by looking at any attribute to differentiate datasets: There will always be another dataset, that is entirely the same as the background set if you are looking at this attribute only. Also there will even be a dataset, that has an attribute that looks more than the background than the actual background<note looks the same as the background but with lower width>. You could say, that finding all alternative datasets, is more about defining your background, than about finding differences. But as even oneOff networks, that are designed to define your background datasets, in theory are expected to be able to be trapped in some features<note see appendix <ref impro>>, the only way to truly evaluate an algorithm, is experimentally. And since we cannot generate all alternative datasets and we have to work with comparing two datasets to each other. But we can at least give some sence of generality to the networks, by looking at different kinds of datasets.
+You could see this, as changing our initial task: Instead of finding one specific anomaly, we now want to find every other anomaly.
+One should notice the huge difference in complexity of this task: Defining every alternative dataset as signals is not solved by looking at any attribute to differentiate datasets: There will always be another dataset, that is entirely the same as the background set, if you are looking at this attribute only. Also there will even be a dataset, that has an attribute that looks more than the background than the actual background<note looks the same as the background but with lower width>. You could say, that finding all alternative datasets, is more about defining your background, than about finding differences. But as even oneoff networks, that are designed to define your background datasets, in theory are expected to be able to be trapped in some features<note see appendix <ref impro>>, the only way to truly evaluate an algorithm, is experimentally. And since we cannot generate all alternative datasets and we have to work with comparing two datasets to each other. But we can at least give some sence of generality to the networks, by looking at different kinds of datasets.
 
 
diff --git a/data/07otherdata/02ldm b/data/07otherdata/02ldm
index 502a38540eddefdd0c349bd898477a8a52cfed4e..e31fab8f6665e70c9febcbefaa3f697503643253 100644
--- a/data/07otherdata/02ldm
+++ b/data/07otherdata/02ldm
@@ -1,7 +1,7 @@
 <subsection title="Ligth dark matter" label="ldm">
 
-This set of datapoints is generated by thorben finke and consists out of jets of transverse momentum between #150*GeV# and #270*Gev# of either QCD jets, or those initiated by a dark matter candidate sugested in <cite ldm>.
-This dataset implies a unsupervised classification task that is way more difficult than the usual top tagging, and as we will see, even more complicated than the other datasets that we test our algorithm on here. <ignore>One reason for this migth be, that the usual top tagging dataset is a very clean, and thus easy to train on, but this does not explain why even other datasets, that is not cleaned up at all are also easier to differentiate. In fact, we thougth multiple times, that there is just a mislabeling in the events, or some other error in the data generation. That beeing said, even though we cannot ever exclude this possibility, we dont think this is the case anymore, and even if it would be the case, this would be the same complex (and thus interresting) task of differentiating neirly same datapoints, and a chance at showing of the benefits of our algorithm.</ignore>
+This set of datapoints is generated by thorben finke and consists out of jets of transverse momentum between #150*GeV# and #270*GeV# of either QCD jets, or those initiated by a dark matter candidate sugested in <cite ldm> (ldm data).
+This dataset implies a unsupervised classification task that is way more difficult than the usual top tagging, and as we will see, even more complicated than the other datasets that we test our algorithm on here. <ignore>One reason for this migth be, that the usual top tagging dataset is a very clean, and thus easy to train on, but this does not explain why even other datasets, that is not cleaned up at all are also easier to differentiate. In fact, we thougth multiple times, that there is just a mislabeling in the events, or some other error in the data generation. That beeing said, even though we cannot ever exclude this possibility, we don`t think this is the case anymore, and even if it would be the case, this would be the same complex (and thus interresting) task of differentiating neirly same datapoints, and a change at showing of the benefits of our algorithm.</ignore>
 The first thing that makes this dataset so much more complicated is the angular distribution: while you can use this distribution to differentiate top jets from their QCD counterparts alone, and this quite well (see chapter <ref simplicity>), here both angular distributions are basically the same
 <i f="angulardistLDM" f2="altangulardistLDM" wmode="True">angular distribution of ldm jets, on the left as 2d histogram and on the rigth as 2 1d histograms (THE SECOND PEAK IS JUST NUMERICS, THAT I WILL STILL FILTER OUT)(i guess newdata/imgs und both (alt)angular)</i>
 and the momentum distribution is not much better
@@ -10,27 +10,27 @@ and the momentum distribution is not much better
 That beeing said, there is one easily understandable parameter that can be used to differentiate both datasets: The number of particles in the jet
 <i f="nhistldm" wmode="True" wid="0.8"> Number size distribution of ldm jets (i guess nedata3/histn.py)</i>
 
-Sadly, this parameter is not very useful because of two reasons
+Sadly, this parameter is not very useful because of two reasons:
 <list>
 <ignore><e> This is not a parameter an image based network has easy access to. You can use special preprocessing to focus on this parameter, but in general that is not a parameter to be easily read out of an image. You migth ask why this matters, since graph based networks (or dense networks for that manner), can access this parameter, possibly showing a clear benefit of graph based networks over image based ones, but</e></ignore>
-<e>Our graph based networks, especcially the ones we talk about in this chapter only have 4 particles. That means that the number of particles just does not enter the network at all<note to be precise, there are o(10) jets with less than 4 particles in our dataset, so it actually is inputted, but only to a neglicible amount>, and even though we can increase the number of particles that enter the network, this will result in less well trained networks (see chapter <ref scale3>), and at the end</i>
-<e>We actually dont want to have a network that just focusses on the number of nodes, as this is a fairly weak way of differentiating jets, resulting only in #O(0.6)# Auc values in the best case. And even though this is way better than any classification score that we achieve in the following, focussing on only one parameter looses every sence of generality</e>
+<e>Our graph based networks, especcially the ones we talk about in this chapter only have 4 particles. That means that the number of particles just does not enter the network at all<note to be precise, there are o(10) jets with less than 4 particles in our dataset, so it actually is inputted, but only to a neglicible amount>, and even though we can increase the number of particles that enter the network, this will result in less well trained networks (see chapter <ref scale3>), and at the end</e>
+<e>We actually don`t want to have a network that just focusses on the number of nodes, as this is a fairly weak way of differentiating jets, resulting only in #O(0.6)# Auc values in the best case. And even though this is way better than any classification score that we achieve in the following, focussing on only one parameter looses every sence of generality</e>
 
 </list>
 
-This means, that we train 4 node networks<note the low number of particles becomes a benefit, since we can be sure not to use the particle number>, that hopefully find some sense of substructre, that makes it possible to differentiate those jets. Sadly this means also, that every result has to be fairly bad! Why? The problem is generality: There is a dataset with different substructure, but there is also another dataset with completely different angular distribution, and since we want our networks to find both, this also means, that having the same angular distribution has some effect on the network making it more likely that this is actually the same datatyp. There migth be still some different substructure, but what would you give more importance to?<ignore> Something that is easily accessible, or something that actually requires a complicated calculation to understand? From a scientific standpoint we would say the easy accessible variable should be more important, but we can do this even a bit better, as we choose variables according to their inverse quadratic relative uncertainity, so what variable is more important? </ignore>There are three effects here
+This means, that we train 4 node networks<note the low number of particles becomes a benefit, since we can be sure not to use the particle number>, that hopefully find some sense of substructre, that makes it possible to differentiate those jets. Sadly this means also, that every result has to be fairly bad<ignore>! Why? The problem is generality</ignore>: There is a dataset with different substructure, but there is also another dataset with completely different angular distribution, and since we want our networks to find both, this also means, that having the same angular distribution has some effect on the network making it more likely that this is actually the same datatyp. There migth be still some different substructure, but<ignore>but what would you give more importance to?</ignore><ignore> Something that is easily accessible, or something that actually requires a complicated calculation to understand? From a scientific standpoint we would say the easy accessible variable should be more important, but we can do this even a bit better, as we choose variables according to their inverse quadratic relative uncertainity, so what variable is more important? </ignore>there are three effects here:
 <list>
-<e>your first thougth migth be, that this relative uncertainity is not affected by the function complexity, and it is true, that if you got only two variables, their effect is completely unaffected by it, but there is a catch</e>
-<e>first, you cannot assume each variable to be exactly known, and a sligth variation in a complex formula can have a much bigger effect than in an easy formula (thing of a momentum 4 vector representing an electron: the formula getting the energy from this 4 vector is much more stable under variations of the 4 momentum, than the formula getting its mass)</e>
-<e>Secondly, the number of neurons is finite, and you could argue, that the number of calculatable variables is not. This means that the network has to choose favorites. And since every given feature can be weigthedly added in a way that (at least for a tiny amount) improves the current relative uncertainity, choosing a complicated/expensive feature also means, not choosing more less complicated features</e>
+<e>Your first thougth migth be, that this relative uncertainity is not affected by the function complexity, and it is true, that if you got only two variables, their effect is completely unaffected by it, but there is a catch.</e>
+<e>You cannot assume each variable to be exactly known, and a sligth variation in a complex formula can have a much bigger effect than in an easy formula (thing of a momentum 4 vector representing an electron: the formula getting the energy from this 4 vector is much more stable under variations of the 4 momentum, than the formula getting its mass).</e>
+<e>Also the number of neurons is finite, and you could argue, that the number of calculatable variables is not. This means that the network has to choose favorites. And since every given feature can be weigthedly added in a way that (at least for a tiny amount) improves the current relative uncertainity, choosing a complicated/expensive feature also means, not choosing more less complicated features.</e>
 
 
 
 </list>
 
-This means, that there is a sligth preference of oneOff networks, to choose easier features<note in general, this is actually a really good thing: Not only does is this statistically useful, but this also means, that oneOffs have a build in regulator, that prevents them from overfitting (at least to a degree), and thus they are a bit more general>, which means, that this is a really hard test for them, and the only thing that we can realistically demand here is invertibility: A Network, trained on ligth QCD jets, that thinks ldm jets are more complicated as well as the inverse.
+This means, that there is a sligth preference of oneoff networks, to choose easier features<note in general, this is actually a really good thing: Not only does is this statistically useful, but this also means, that oneoffs have a build in regulator, that prevents them from overfitting (at least to a degree), and thus they are a bit more general>, which means, that this is a really hard test for them, and the only thing that we can realistically demand here is invertibility: A Network, trained on ligth QCD jets, that thinks ldm jets are more complicated as well as the inverse.
 <i f="multisep10" wmode="True">ldm jet invertibility (multisep.py)</i>
-As you see, this is not at all trivial, but when we consider the loss of the oneOff network, which is drawn on the x axis as quality, the best networks are actually invertible. And since this is still completely unsupervised, just using the feature quality of a network, we can say that we can generate invertible anomaly detection algorithms on this dataset. 
+As you see, this is not at all trivial, but when we consider the loss of the oneoff network, which is drawn on the x axis as quality, the best networks are actually invertible. And since this is still completely unsupervised, just using the feature quality of a network, we can say that we can generate invertible anomaly detection algorithms on this dataset. 
 That beeing said, this is obviously not useful at all: half a percentage in AUC does not help you at differentiating new physics, but it is worth to note, that also top tagging after normalization looked very similar to this once (see chapter <ref invertibility>), and seeing that there the classification quality improved a lot, we see no reason, why you could not improve and optimize this network to be drastically better. Especcially, since we did not run any hyperparameter optimization (except for the compression size, which is 1 bigger (at 10)<note we think, that this higher compression size allows the network to understand more subfeatures>), and still only use 4 particles<ignore>, which even when using supervised algorithm is not easy to differentiate
 <i f="none" wmode="True">(not yet having a final version)supervised 4 particle ldm</i></ignore>
 
diff --git a/data/07otherdata/03otherdata b/data/07otherdata/03otherdata
index c88462d01bd28ae2a9474ba7e9d2a9fbb6d830bd..8ebced365d3096535d826bae2c9bb2d206d62569 100644
--- a/data/07otherdata/03otherdata
+++ b/data/07otherdata/03otherdata
@@ -1,17 +1,17 @@
 <subsection title="Other datasets" label="otherdata">
 
 
-<subsubsection title="Quark v gluon" label="qg">
+<subsubsection title="Quark or gluon" label="qg">
 
-Quark and gluon data, is generated by madgraph<cite madgraph>, Pythia<cite pythia> and delphes<cite delphes>. One set is generated as parton parton to gluon gluon collisions and another as parton parton to two parton without gluon collisions. Jets are used, if there transverse jet momentum is between 550 and 650 geV. This data was used originally to see if a QCD trained classifier makes a easily accessible difference between quarks and gluons<note you could interpret this, as another form of complexity: while top jets are all the result of top quarks, with QCD jets there are multiple options, we though this could explain why QCD trained encoder are generally worse, but this just not the case>, but even though this is seems not to be the case, we can still use this dataset to test our algorithm a bit further. Again we use 4 particle networks, with a compression size of #9# and only neglicible hyperparameter optimization to reach quality of
+Quark and gluon data, is generated by madgraph<cite madgraph>, Pythia<cite pythia> and delphes<cite delphes>. One set is generated as parton parton to gluon gluon collisions and another as parton parton to two parton without gluon collisions. Jets are used, if there transverse jet momentum is between 550 and 650 geV. This data was used originally to see if a QCD trained classifier makes a easily accessible difference between quarks and gluons<note you could interpret this, as another form of complexity: while top jets are all the result of top quarks, with QCD jets there are multiple options, we though this could explain why QCD trained encoder are generally worse, but this just not the case>, but even though this is seems not to be the case, we can still use this dataset to test our algorithm a bit further. Again we use 4 particle networks, with a compression size of #9# and only neglicible hyperparameter optimization to reach quality of sligthly above random.
 <i f="drquarkgluon" f2="dsquarkgluon" wmode="True">Roc curves for quark gluon</i>
 As you see, these are invertible networks, and even though they are not very good ones, as described in the previous chapter <ref ldm> this does not really matter, since optimization has the potential to improve them quite a lot. <cite quarkdata> could be seen as a reference paper for this process, even though they use a supervised approach and high level input data on different transverse momentum ranges, their achieves AUC values below #0.9# suggest that this tagging job is more complicated than the usual top tagging. Also chapter <ref crossdata> will support this hypothesis. 
 
-<subsubsection title="leptons" label="leptons">
+<subsubsection title="Leptons" label="leptons">
 
 
 
-This dataset is not very physically useful, and more interresting from an anomaly detection standpoint: We again generate particle collisions using madgraph, Pythia and delphes, but instead of partons colliding into partons, we use leptons colliding and producing partons. For the first set, we use any combination of electrons and muons with arbitrary charge, and for the second one we only use tau leptons. We also use a fairly big transverse momentum range for the jet of #20*Gev# to #5000*Gev# to see if our algorithm is affected by this bigger range.
+This dataset is not very physically useful, and more interresting from an anomaly detection standpoint: We again generate particle collisions using madgraph, Pythia and delphes, but instead of partons colliding into partons, we use leptons colliding and producing partons. For the first set, we use any combination of electrons and muons with arbitrary charge, and for the second one we only use tau leptons. We also use a fairly big transverse momentum range for the jet of #20*GeV# to #5000*GeV# to see if our algorithm is affected by this bigger range.
 
 <i f="drleptons" f2="dsleptons" wmode="True">Roc curves for leptons</i>
 
diff --git a/data/07otherdata/05cross b/data/07otherdata/05cross
index b9bbb07e23cc624c2f24330a182cf7dc35c88a2a..02a888e2577a3d6d1cb19e8c3d33b98ee111c021 100644
--- a/data/07otherdata/05cross
+++ b/data/07otherdata/05cross
@@ -1,9 +1,9 @@
-<subsection title="Even more comparison" label="crossdata">
+<subsection title="Cross comparisons" label="crossdata">
 
-At the beginning of this chapter, we called anomaly detection the task of finding everything that is not similar to the trained on class. And even though we tried to evaluate this task, by showing the invertibility on a multitude of datasets, we slowly are out of particles to test it on<note especcially since the initial toptagging dataset already contained the whole of QCD>. That beeing said, one thing we did not yet do, is to mix the datasets. You migth question how useful this is from a physical standpoint, as there will probably never be a situation, in which you want to find leptons, only knowing gluons. The point is that new physics could have a neirly arbitrary form, and even though we will never live in world in which we only know about gluons, finding data that does not look like gluons is very useful. We think that these experiments introduce thrust in the algorithm used, as chapter <ref secinv> clearly shows, that invertibility and feature triviality can be linked. <ignore>Also from a computational standpoint, for a QCD trained network, also tau generated jets are different,</ignore> And since training unsupervised does even mean, that we dont have to train new anomaly detection models, there is no reason not to compare those jets
+At the beginning of this chapter, we called anomaly detection the task of finding everything that is not similar to the trained on class. And even though we tried to evaluate this task, by showing the invertibility on a multitude of datasets, we slowly are out of particles to test it on<note especcially since the initial toptagging dataset already contained the whole of QCD>. That beeing said, one thing we did not yet do, is to mix the datasets. You migth question how useful this is from a physical standpoint, as there will probably never be a situation, in which you want to find leptons, only knowing gluons. The point is that new physics could have a neirly arbitrary form, and even though we will never live in world in which we only know about gluons, finding data that does not look like gluons is very useful. We think that these experiments introduce thrust in the algorithm used, as chapter <ref secinv> clearly shows, that invertibility and feature triviality can be linked. <ignore>Also from a computational standpoint, for a QCD trained network, also tau generated jets are different,</ignore> And since training unsupervised does even mean, that we don`t have to train new anomaly detection models, there is no reason not to compare those jets.
 <i f="crosssep" wmode="True"> cross invertibility (crosssep)</i>
 
-As you see, there are only few spots that are not invertible (we chanced the meaning of each AUC value, in such a way, that each slot should be deeply blue in the best case). For simplicity we mark the noninvertible networks with black spots, but this still does not allow to see every value, so here are those comparisons again as a table
+As you see, there are only few spots that are not invertible (we changed the meaning of each AUC value, in such a way, that each slot should be deeply blue in the best case). For simplicity we mark the noninvertible networks with black spots, but this still does not allow to see every value, so here are those comparisons again as a table.
 
 
 
@@ -25,9 +25,9 @@ As you see, there are only few spots that are not invertible (we chanced the mea
 
 </table>
 
-As you see, everything is either invertible, or at least very close. Furthermore, only 4 noninvertible comparisons exist, and are always less than #2.5#\% worse than random guessing, and are trained on ligth dark matter or ligth QCD data, which as explained in <ref ldm> is hard to differentiate<note it is a bit weird, that ligth QCD jets are more similar to ligth dark matter data, than to QCD of higher energy, especially since every value is normated quite thorougly (chapter <ref normalization>), but this still does not mean, that their is something wrong with the data, as the normalization has no effect on they number of particles, but just on the size of each value. And since higher energy jets can decay differently, this migth explain why ligth QCD and ligth dark matter jets look so similar >.
+As you see, everything is either invertible, or at least very close. Furthermore, only 4 noninvertible comparisons exist, and are always less than #2.5#\% worse than random guessing, and are trained on ligth dark matter or ligth QCD data, which as explained in <ref ldm> is hard to differentiate<note it is a bit weird, that ligth QCD jets are more similar to ligth dark matter data, than to QCD of higher energy, especially since every value is normalized quite thorougly (chapter <ref normalization>), but this still does not mean, that their is something wrong with the data, as the normalization has no effect on they number of particles, but just on the size of each value. And since higher energy jets can decay differently, this migth explain why ligth QCD and ligth dark matter jets look so similar >.
 
-Also please note, that the rows and collumns that are related to top jets are clearly visible: It seems to be a much easier task to differentiate top jets, than every other dataset, even when we use normated networks. This suggests that only using top jets as anomalies artificially inflates your performance.
+Also note, that the rows and collumns that are related to top jets are clearly visible: It seems to be a much easier task to differentiate top jets, than every other dataset, even when we use normalized networks. This suggests that only using top jets as anomalies artificially inflates your performance.
 
 
 
diff --git a/data/09fazit/08future b/data/09fazit/08future
index 40f801a2d89c1fe1e91db2752dc663c034bf3652..06f13fd30f60bafa225610b023fc0a6c7c3598e9 100644
--- a/data/09fazit/08future
+++ b/data/09fazit/08future
@@ -8,7 +8,7 @@ THNKING ABOUT SOMETHING LIKE ABOUT WRITING STUFF JUST DOES NOT WORK, THIS LIST W
   <e>Vary losses</e>
   <l2st>
     <e>Actual image like losses (calculate histograms and compare them)</e>
-    <e>permutation invariant losses, we already tried some, but they dont yet work well, even though they theoretically should</e>
+    <e>permutation invariant losses, we already tried some, but they don`t yet work well, even though they theoretically should</e>
     <e>L1.5</e>
     <e>Image like losses</e>
     <l3st>
@@ -19,7 +19,7 @@ THNKING ABOUT SOMETHING LIKE ABOUT WRITING STUFF JUST DOES NOT WORK, THIS LIST W
   </l2st>
   <e>Other data</e>
   <l2st>
-    <e>It seems to be easier to work on transformed 4 momenta, but on classical 4 vectors, explain and chance this</e>
+    <e>It seems to be easier to work on transformed 4 momenta, but on classical 4 vectors, explain and change this</e>
     <e>add lowly correlated variables like the mass</e>
     <e>see if you can use autoencoder to test assumptions about symmetry (like the position of the jet in the detector does not matter)</e>
   </l2st>
@@ -43,9 +43,9 @@ THNKING ABOUT SOMETHING LIKE ABOUT WRITING STUFF JUST DOES NOT WORK, THIS LIST W
   <e>Scaling</e>
   <l2st>
     <e>no more nans</e>
-    <e>improve networks that dont use a latent space of just one node</e>
+    <e>improve networks that don`t use a latent space of just one node</e>
   </l2st>
-  <e>Use other scores then AUCs, and try to optimize oneOff networks for them</e>
+  <e>Use other scores then AUCs, and try to optimize oneoff networks for them</e>
   <e>explain why better compression algorithm work worse</e>
   <e>explain the variance in AUCs you get by using better decompression</e>
   
diff --git a/data/10anhang/01failed b/data/10anhang/01failed
index 985cbae553ab1b2c6adf47b47c9c86f12289d6b9..4047ef64e03abef5bae6ff1ccf885fb921860748 100644
--- a/data/10anhang/01failed
+++ b/data/10anhang/01failed
@@ -6,7 +6,7 @@ In this chapter we will quickly go over some bad ideas you could have, on how to
 These implementations are usually defined by an encoding and a decoding algorithm, so basically something to go from a big graph to a small graph, and something to reverse this again. In addition to this, the graph update and the graph construction stay mostly the same as it was explained in chapter <ref gnn> and <ref imgsetups>.
 
 <subsubsection title="trivial models" label="failedtrivial">
-Let us start with the probably most simple autoencoder algorithms: To make a #n# node graph into a #m# node graph, we just cut away the last nodes until there are only #m# nodes left<note please note the importance of the #p_T# ordering here: Cutting the last particles means cutting the particles with lowest #p_T# and thus the probably least important particles> to reduce the graph size, and add zero valued particles to it again. One difficulty here lies in the fact that those particles have no more graph connections, this we solved by just keeping the original graph connections stored. Sadly, those networks just dont work: even when we would set the compression size over the input size, the reproduced jets hardly bare any resemble to the input jets: This is the first example of the central problem of graph autoencoding: Permutation invariance. Consider the following encoder: two numbers #a# and #b# where #Eq(a,b+1)#, this would be trivial to compress into one number for a normal<note dense> Autoencoder(maybe just take #a#), but here we have to respect permutaion symmetry, so basically we do not know what the first and what the second particle is and how do we decompress now? In this context you could keep one of the parameters and try to encode if the other one is bigger or smaller than this, maybe you also know that #LessThan(0,a)# and you could multiply it by #-1# if it is the smaller one, but this is less than trivial, and by increasing the number of parameters this gets even more complicated. This is a problem that mostly appears as the inability of even a "good" Autoencoder to work with and compression size that is equal to the input size, building an identity (see chapter <ref identities>). <ignore>Next to the loss from the compression, there seems to still be a certain loss from the graph structure, given at least partially coming from permutation invariance.</ignore>
+Let us start with the probably most simple autoencoder algorithms: To make a #n# node graph into a #m# node graph, we just cut away the last nodes until there are only #m# nodes left<note please note the importance of the #p_T# ordering here: Cutting the last particles means cutting the particles with lowest #p_T# and thus the probably least important particles> to reduce the graph size, and add zero valued particles to it again. One difficulty here lies in the fact that those particles have no more graph connections, this we solved by just keeping the original graph connections stored. Sadly, those networks just don`t work: even when we would set the compression size over the input size, the reproduced jets hardly bare any resemble to the input jets: This is the first example of the central problem of graph autoencoding: Permutation invariance. Consider the following encoder: two numbers #a# and #b# where #Eq(a,b+1)#, this would be trivial to compress into one number for a normal<note dense> Autoencoder(maybe just take #a#), but here we have to respect permutaion symmetry, so basically we do not know what the first and what the second particle is and how do we decompress now? In this context you could keep one of the parameters and try to encode if the other one is bigger or smaller than this, maybe you also know that #LessThan(0,a)# and you could multiply it by #-1# if it is the smaller one, but this is less than trivial, and by increasing the number of parameters this gets even more complicated. This is a problem that mostly appears as the inability of even a "good" Autoencoder to work with and compression size that is equal to the input size, building an identity (see chapter <ref identities>). <ignore>Next to the loss from the compression, there seems to still be a certain loss from the graph structure, given at least partially coming from permutation invariance.</ignore>
 That beeing said, permutation invariance can also be a benefit, especially in permutation invariant input data, more to this in chapter <ref aperminv>
 
 
diff --git a/data/10anhang/02diffalgo b/data/10anhang/02diffalgo
index e35eb8ebfc07e1f5a179bea1a9b62cfc0ef2ce77..fc1fbefe0a9a95ca93523c649cc423376bb159a6 100644
--- a/data/10anhang/02diffalgo
+++ b/data/10anhang/02diffalgo
@@ -15,7 +15,7 @@ K neirest neighbour is the first algorithm that improves over simply using the a
 
 
 <subsubsection title="oneoff" label="mixedoo">
-Oneoffs seem to be the way to go here, and will be used exclusively for the rest of this chapter: One network reached about #0.247# with an error of #0.005#, already beating all all our autoencoder, and by combining multiple ones, you can reach an AUC of about #0.2# beeing quite good.
+oneoffs seem to be the way to go here, and will be used exclusively for the rest of this chapter: One network reached about #0.247# with an error of #0.005#, already beating all all our autoencoder, and by combining multiple ones, you can reach an AUC of about #0.2# beeing quite good.
 
 
 <i f="none" wmode="True">(will be written later)comparison of all one class algos</i>
diff --git a/data/10anhang/02simpleae b/data/10anhang/02simpleae
index 5ca9269acb3719dd09e09532e14d6ceeb36b029d..5567178a814457ec7a939cf2e5c1b2f54d800232 100644
--- a/data/10anhang/02simpleae
+++ b/data/10anhang/02simpleae
@@ -7,7 +7,7 @@ As you see, this autoencoder takes QCD jets, transforms them as introduced in <r
 
 
 <subsubsection title="topK" label="quicktopK">
-The probably most comonly used algorithm, to construct a set of graph connections from a list of vectors, topK, seems to be quite easy to understand: you connect each vector, to the #K# vectors that are most similar to it. The difficulty lies in the word similar: Here two vectors are more similar, the smaller the l2 difference is. In an attempt, to make this more powerful, we also use a learnable metrik in this l2 difference. Even though this migth not be strictly neccesary, since the network can chance parameters to accomodate its sence of similarity, this still allows the network to better choose what to focus on in each topK layer. It can be quite useful for autoencoder, since for example ignoring a parameter, could else only be done, by decreasing its size in relation to the other parameters, which migth not be optimal, when you want an accurate reproduction. This also allows you to create a graph, before having any learnable layers. On the other hand, these metrik can complicate the calculcation of the adjacency matrix, which we try to manage by demanding that the metrik is entirely diagonal, reducing also the needed time drastically, and the parameters of the metrik can increase the occurence of divergences in training, since even a small chance of those parameters can effect the network output in huge ways. That beeing said, having a humanly understandable metrik, can lead to interresting insigths (see appendix <ref ametrikana>).
+The probably most comonly used algorithm, to construct a set of graph connections from a list of vectors, topK, seems to be quite easy to understand: you connect each vector, to the #K# vectors that are most similar to it. The difficulty lies in the word similar: Here two vectors are more similar, the smaller the l2 difference is. In an attempt, to make this more powerful, we also use a learnable metrik in this l2 difference. Even though this migth not be strictly neccesary, since the network can change parameters to accomodate its sence of similarity, this still allows the network to better choose what to focus on in each topK layer. It can be quite useful for autoencoder, since for example ignoring a parameter, could else only be done, by decreasing its size in relation to the other parameters, which migth not be optimal, when you want an accurate reproduction. This also allows you to create a graph, before having any learnable layers. On the other hand, these metrik can complicate the calculcation of the adjacency matrix, which we try to manage by demanding that the metrik is entirely diagonal, reducing also the needed time drastically, and the parameters of the metrik can increase the occurence of divergences in training, since even a small change of those parameters can effect the network output in huge ways. That beeing said, having a humanly understandable metrik, can lead to interresting insigths (see appendix <ref ametrikana>).
 You could ask yourself, if a topK algorithm is the best choice, since the number of possible adjacency matrices is quite low, see for this appendix <ref atopkwhy>.
 Finally, it should be noted, that the topK layer can increase the size of each of the feature vectors, which is useful for the compression algorithm, even though in this specific example this is not used.
 
@@ -31,7 +31,7 @@ Since the compression stage reordered each feature vector, while the decompressi
 <subsubsection title="Training setup" label="quicktrain">
 Another thing that has to be clearified concerning this model, is the training procedure. We use the adam optimizer, with a learning rate of #0.001#, with a batch size of #200# and train the network, using an earlyStopping callback, until it does no longer improve its validation loss for #10# epochs and afterwards use the epoch with the minimal validation loss. We use #600000# training and #200000# validation jets to plot here the loss for each epoch
 <i f="historyb00" wmode="True">training for b1/00</i>
-As you see, there is not really any progress made in the training<note except for maybe the first epoch, which is not shown in these kind of plots>, but you already see one fact, that will be quite common in the following: The validation loss is not (much) bigger than the training loss, neither at the end, not anywhere. This is fairly uncommon, as usually earlyStopping is used to compat overfitting, and validation losses that seem to increase at some point, but also easily explained, since encoder and decoder only amount to a total of 840 trainable parameters, which is not enough to store informations for #O(100000)# events. Interrestingly, this seems to be a clear benefit for graph autoencoder, as even bigger networks with similar amounts of parameters, trained on less data, dont seem to show any tendency to overfit. This allows us to reduce the training size to at least 2 orders of magnitude less, without any quality loss (see chapter <ref asize>), and you could even ask yourself if it would not be possible to remove the whole need of splitting your data into training and validation data. That beeing said, this dataseperation is mentained for the rest of the thesis, and this overfitting safety comes at a price: the validation loss migth not increase in relation to the training loss, but that does not mean that both cannot increase in parallel. This, and the fact that graph training curves are way more noisy than usual training curves, make earlyStopping still a viable training callback, and result in most of the reasons, each training stops.
+As you see, there is not really any progress made in the training<note except for maybe the first epoch, which is not shown in these kind of plots>, but you already see one fact, that will be quite common in the following: The validation loss is not (much) bigger than the training loss, neither at the end, not anywhere. This is fairly uncommon, as usually earlyStopping is used to compat overfitting, and validation losses that seem to increase at some point, but also easily explained, since encoder and decoder only amount to a total of 840 trainable parameters, which is not enough to store informations for #O(100000)# events. Interrestingly, this seems to be a clear benefit for graph autoencoder, as even bigger networks with similar amounts of parameters, trained on less data, don`t seem to show any tendency to overfit. This allows us to reduce the training size to at least 2 orders of magnitude less, without any quality loss (see chapter <ref asize>), and you could even ask yourself if it would not be possible to remove the whole need of splitting your data into training and validation data. That beeing said, this dataseperation is mentained for the rest of the thesis, and this overfitting safety comes at a price: the validation loss migth not increase in relation to the training loss, but that does not mean that both cannot increase in parallel. This, and the fact that graph training curves are way more noisy than usual training curves, make earlyStopping still a viable training callback, and result in most of the reasons, each training stops.
 
 
 <subsubsection title="Results" label="quickres1">
@@ -77,7 +77,7 @@ So the encoder could be described as: First, read and preprocess your data, do a
 If we talk about decision quality, this network is not great, but also not terrible
 <i f="00_imgs_recqual" f2="00_imgs_roc" wmode="True">Two images, once showing the recqual of b1.0, and the other one showing roc of b1.0</i>
 
-The more interresting question is the question of quality. The images i show here show #mean(abs(f))#<note this is done, since for angular features my preprocessing sets the mean to zero, and i dont want my information to cancel themself> and the standart deviation of one input and output feature over each other. 
+The more interresting question is the question of quality. The images i show here show #mean(abs(f))#<note this is done, since for angular features my preprocessing sets the mean to zero, and i don`t want my information to cancel themself> and the standart deviation of one input and output feature over each other. 
 
 The first interresting is the momentum plot
 <i f="00_imgs_outputqual3" wmode="True">pt outputqual plot b1.0</i>
@@ -88,7 +88,7 @@ This looks (????????)
 
 The alternative is given by the #phi# plot (the #eta# plot looks basically the same)
 <i f="00_imgs_outputqual2" wmode="True">phi outputqual plot b1.0</i>
-As you see, the mean values dont quite match, these mean values are consistently reproduced below the input mean values (This i explain in chapters <ref imgout> and <ref losses>). and also the standart deviations are way smaller reproduced. So basically, this is not what the network cares about<note you can controll the focus of the network by chancing the normalization of the input data>, which is a problem, that will become interresting in chapter <ref secinv>.
+As you see, the mean values don`t quite match, these mean values are consistently reproduced below the input mean values (This i explain in chapters <ref imgout> and <ref losses>). and also the standart deviations are way smaller reproduced. So basically, this is not what the network cares about<note you can controll the focus of the network by changing the normalization of the input data>, which is a problem, that will become interresting in chapter <ref secinv>.
 
 Finally there is flag:
 <i f="00_imgs_outputqual0" wmode="True">flag outputqual of b1.0</i>
diff --git a/data/10anhang/03goodae b/data/10anhang/03goodae
index be3ec80bc1d321e1123c281b95eb635c0ff817a6..581f36f52486b0587cefe093202a6ac3bbbd4489 100644
--- a/data/10anhang/03goodae
+++ b/data/10anhang/03goodae
@@ -1,6 +1,6 @@
 <subsection title="Improving autoencoder" label="secondworking">
 
-Given the fairly good AUC score, it looks like to only thing we now need to do, is to increase the size of this autoencoder, and we probably have a really great anomaly detection algorithm. But before we try, and fail<note see chapter <ref scaling>>, at this, let us improve our autoencoder first. As you migth agree, the training curve does not look very impressive, and the reconstruction is also not very good. Thats why we suggest some chanced model<note we alter models iteratively, but since we dont want to show tausends of models here, you only see summaries, which is why the changes seem a bit random>.
+Given the fairly good AUC score, it looks like to only thing we now need to do, is to increase the size of this autoencoder, and we probably have a really great anomaly detection algorithm. But before we try, and fail<note see chapter <ref scaling>>, at this, let us improve our autoencoder first. As you migth agree, the training curve does not look very impressive, and the reconstruction is also not very good. Thats why we suggest some changed model<note we alter models iteratively, but since we don`t want to show tausends of models here, you only see summaries, which is why the changes seem a bit random>.
 
 <i f="vis200" wmode="True">Network setup plot(c1/200)</i>
 
diff --git a/data/10anhang/03optimalae b/data/10anhang/03optimalae
index 49927b4c5dc17826980638ec5553cb1495e7b676..ab420dbdab589ad41bf833d95c63a589ddf5bb03 100644
--- a/data/10anhang/03optimalae
+++ b/data/10anhang/03optimalae
@@ -3,11 +3,11 @@
 There are some algorithmical changes that we thougth of, that will be testet in this chapter.
 
 <subsubsection title="Physical intuition behind the encoding algorithm" label="intuitivecode">
-The usual encoding algorithm could be seen, as inverting a particle decay: Taking for example a simple two particle decay: On the graph, you could understand it as some function, which is making 1 node into two nodes. And as you can find the original particle by some function of the resulting particles, you can use an original particle with some additional attributes<note like what it decays into (also ignoring uncertainities for now)> to reconstruct the new particles from it. This migth suggest that this kind of autoencoder is optimal for particle physics, but this setup is even more useful as it does not simply cut away additional information, and the physical problem is actually not that optimal, since the number of particles in each decay does not only have to be constant, but also known before in every compression step. Also particles dont decay in steps: It could well be, that the initial particle decays into two particles, of which only one constinuous to decay further. That beeing said, the optimal encoding algorithm, that we would like to be able to write (appendix <ref aultcode>), would be solve this, and thus have even more physical intuition. 
+The usual encoding algorithm could be seen, as inverting a particle decay: Taking for example a simple two particle decay: On the graph, you could understand it as some function, which is making 1 node into two nodes. And as you can find the original particle by some function of the resulting particles, you can use an original particle with some additional attributes<note like what it decays into (also ignoring uncertainities for now)> to reconstruct the new particles from it. This migth suggest that this kind of autoencoder is optimal for particle physics, but this setup is even more useful as it does not simply cut away additional information, and the physical problem is actually not that optimal, since the number of particles in each decay does not only have to be constant, but also known before in every compression step. Also particles don`t decay in steps: It could well be, that the initial particle decays into two particles, of which only one constinuous to decay further. That beeing said, the optimal encoding algorithm, that we would like to be able to write (appendix <ref aultcode>), would be solve this, and thus have even more physical intuition. 
 
 <subsubsection title="better encoding" label="encoding">
 Since writing this much more advanced graph abstraction algorithm, would have taken very much time, let us focus first on a bit more simple better encoding algorithm:
-The current encoding basically completely ignores any graph information. After any compression stage the whole graph has to be relearned, and connections only indirectly<note through the preciding graph update steps> affect the corresponding feature vectors. Why not use the graph a bit more? Here we suggest that using a function of the original graph as the compressed graph migth be a good idea: When compressing #n# vectors, you can see the adjacency matrix as a matrix of matrices, and the only task you need to solve, is how to extract some form of this initial global matrix. This is done here, by applying a function to each submatrix. We try out setting this function to be the mean, the maximum or the minimum of the original connections and compare them with or without rounding each entry to be one or zero to the usual graph compression. With the rounding you can see those options as setting a connection to exist when more original connections exist than dont, when at least one connection exist, or when all connections exist.
+The current encoding basically completely ignores any graph information. After any compression stage the whole graph has to be relearned, and connections only indirectly<note through the preciding graph update steps> affect the corresponding feature vectors. Why not use the graph a bit more? Here we suggest that using a function of the original graph as the compressed graph migth be a good idea: When compressing #n# vectors, you can see the adjacency matrix as a matrix of matrices, and the only task you need to solve, is how to extract some form of this initial global matrix. This is done here, by applying a function to each submatrix. We try out setting this function to be the mean, the maximum or the minimum of the original connections and compare them with or without rounding each entry to be one or zero to the usual graph compression. With the rounding you can see those options as setting a connection to exist when more original connections exist than don`t, when at least one connection exist, or when all connections exist.
 THIS DATA IS STILL IN WORK PARTIALLY
 
 The data we compare it on here, includes all the stuff we implement over the remaining chapters, which is why there is an oneoff auc in those tables(see chapter <ref secmixed>), and also why the quality is generally worse (see chapter <ref secinv>).
@@ -61,11 +61,11 @@ Another thing we test, is how well a learnable parameter transformation, as used
 
 </table>
 
-Evaluating this test series is not as easy as the last one. In the loss, the comparison network is better than all other layers, excluding rounded means with a min function, but the oneoff auc is worse at this model, and at all other ones. That beeing said, when setting the function to be the mean, the usual auc beats the oneoff one a bit. We choose not to use this, because the higher loss means that worse autoencoder produce these results, and a consistent AUC score, that is not reproduced by the oneOff AUC suggests trivial features, which suggest no generality and invertibility.
+Evaluating this test series is not as easy as the last one. In the loss, the comparison network is better than all other layers, excluding rounded means with a min function, but the oneoff auc is worse at this model, and at all other ones. That beeing said, when setting the function to be the mean, the usual auc beats the oneoff one a bit. We choose not to use this, because the higher loss means that worse autoencoder produce these results, and a consistent AUC score, that is not reproduced by the oneoff AUC suggests trivial features, which suggest no generality and invertibility.
 
 
 <subsubsection title="better decoding" label="decoding">
-Also the decoder, does not use the graph structure completely. So we try to replace the abstraction with a constant learnable graph, by an abstraction with a graph that is not constant. The problem here, is that the tensorproduct introduced in <ref identies> and <ref gnn> does not work for a product of one graph with multiple graphs. The main difficulty lies in finding out how to work with the nondiagonal terms: Consider again adjacency matrices of adjacency matrices: When each feature vector becomes a vector of feature vectors, also each entry in the adjacency matrix becomes a new matrix. These matrices, multiplied with the original entry would result in a tensorproduct, when the new matrices would always be the same, but this is what we want to chance. Finding now the diagonal matrices can be left to a learnable function of the feature vector, but for the offdiagonal matrices, we have two suggestions: The first, graphlike decompresser, define those matrices as functions of the two corresponding diagonal matrices. Here we compare a product, a sum and those rounded versions and and or not only to the abstraction with a constant graph, but also to the second suggestion: paramlike decompresser: instead of the diagonal matrices beeing functions of a feature vector, every submatrix is a learnable function of its two corresponding original feature vectors.
+Also the decoder, does not use the graph structure completely. So we try to replace the abstraction with a constant learnable graph, by an abstraction with a graph that is not constant. The problem here, is that the tensorproduct introduced in <ref identies> and <ref gnn> does not work for a product of one graph with multiple graphs. The main difficulty lies in finding out how to work with the nondiagonal terms: Consider again adjacency matrices of adjacency matrices: When each feature vector becomes a vector of feature vectors, also each entry in the adjacency matrix becomes a new matrix. These matrices, multiplied with the original entry would result in a tensorproduct, when the new matrices would always be the same, but this is what we want to change. Finding now the diagonal matrices can be left to a learnable function of the feature vector, but for the offdiagonal matrices, we have two suggestions: The first, graphlike decompresser, define those matrices as functions of the two corresponding diagonal matrices. Here we compare a product, a sum and those rounded versions and and or not only to the abstraction with a constant graph, but also to the second suggestion: paramlike decompresser: instead of the diagonal matrices beeing functions of a feature vector, every submatrix is a learnable function of its two corresponding original feature vectors.
 
 
 <table caption="Quality differences for different graph like decoder" label="decode1" c=6>
diff --git a/data/10anhang/03whygae b/data/10anhang/03whygae
index a48b6da2cd3bfdf7a5b7714c67a3d6cf22da26f3..43690f277f8c955d8b6c4b55d7d589855acfddde 100644
--- a/data/10anhang/03whygae
+++ b/data/10anhang/03whygae
@@ -4,11 +4,11 @@
 NOT AT THE RIGTH POSITION
 </ignore>
 
-From our experience, Graph autoencoder have some clear advantages over classical autoencoder. This does not mean, that they dont have problems(as discussed in chapter <ref whynotgae>), or even that those benefits usually outweigh the problems, but at least that there migth be situations, in which choosing a graph autoencoder would be a good choice.
+From our experience, Graph autoencoder have some clear advantages over classical autoencoder. This does not mean, that they don`t have problems(as discussed in chapter <ref whynotgae>), or even that those benefits usually outweigh the problems, but at least that there migth be situations, in which choosing a graph autoencoder would be a good choice.
 The most obvious situation, in which graph autoencoder should be choosen, is defined by their input data, if it has the form of a graph: That means multiple vectors, with some relational information between them, that should not be ignored. This can also be beneficial for variable number of vectors, or when a permutation symmetry between the inputs is expected.
 Another benefit is the seperation in multiple similarly handled vectors. This similar handling does not only keep the number of trainable parameters low, and thus makes overfitting hard<note or in the models we trained basically impossible>, but also makes interpreting the output easier, since when every attribute of the same kind is treated the same, there are not many differences between the qualities of different particles, but more between different attributes.
 Also and probably most useful, these shared parameters<note the low count and the demanded similarity> keep the number of needed training samples quite low: Even though more training sample cannot really hurt the network quality, we could without problems, reduce the trainingsize down by more than two orders of magnitude from 600k to 5k (see appendix <ref asize>), and it seems to be possible to reduce  these training size oven further, allowing us to build useful networks with only #O(100)# training samples (see chapter <ref feyn>).
-Finally, chancing the graph setup layer can chance the whole meaning of a graph layer, transforming a layer that handles physical distance into one that cares only about momenta. This can allow for variable metrik setups, that can iteratively focus on whatever is important at the current position.
+Finally, changing the graph setup layer can change the whole meaning of a graph layer, transforming a layer that handles physical distance into one that cares only about momenta. This can allow for variable metrik setups, that can iteratively focus on whatever is important at the current position.
 
 <ignore>
 MAYBE SOMETHING ABOUT VARIABLE MEANING OF LOCALITY
diff --git a/data/10anhang/04whyaesuck b/data/10anhang/04whyaesuck
index b475a82d22482275fa1af089033ff254058b5021..f1c33dbff7dea543be718741fa52dec5ce74b403 100644
--- a/data/10anhang/04whyaesuck
+++ b/data/10anhang/04whyaesuck
@@ -8,22 +8,22 @@ Given the reasons for why to use graph autoencoder in chapter <ref whygae>, here
 
 <subsubsection title="Reproduding vs classifing quality" label="nogaequal">
 When we started working on anomaly detection, Autoencoder seemed like are quite a good idea, a simple way to differentiate between things that are known, and things that are not known, while still giving you a way of testing how good your models are trained, without needing anything else but background data by just evalutating the quality of the autoencoder. It stands to reason that a bad autoencoder did not understand the background data in any way, that can be used to differentiate it from the signal data, but this is basically just an assumption, and so after having a bit more experience encoding and classifying data, and especcially when we now have another method of seperating data, we first want to spend some time evaluating this hypothesis.
-To do this, lets look first at the loss of the autoencoder: Since it is basically just the difference between input and output<note simplifying chapter <ref losses> a bit>, it is a measurement about how good the autoencoder reproduces whatever we put into it<note this is also a bit of a simplification, since the normalization of the data matters a lot, but we will come back to this later>, and so by just calculating the loss on background data, we have a measure for the quality of the autoencoder. <ignore>Also, as argued for in the chapter about decicision quality <ref binclass>, we will use the AUC score to evalutate the classifier quality.</ignore> So basically we want to see a strong falling correlation between the loss of our network and the AUC score<note the lower the loss, the higher the AUC, since we train on QCD jets>, do we see this? We show here the loss in training against the decision quality of the corresponding network in this trainingstep. Please note that this is exactly what we want, since the correlation we want here, would mean, that in training, a classifier gets continously better<note yes you could argue, that it is enough when the highest AUC is reached at the lowest loss, but in practice this is not enough because the network does not always reach the same point><note it should be noted, that a network cancels the training when it does not improve for a certain number of epochs, so in theory you do not know if there may not even be a better classifier at a way later epoch, but with worse loss. This is usually assumed to be false, since overfitting defines the rest of training, but since we do not see basically any overfitting, this migth not be so easily ignored><note we should also note, that this is an analysis that we did only for a small fraction of all networks, since evalutating the quality of hunderts of networks takes considerable time, often even more than training itself. This is also why we chance this method later, to consider only those epochs in which the network improves its loss>.
-<i wmode="True" f="none">(FORGOT THIS PLOT, AND CALCULATION TAKES TIME)A simple old AUC by epoch plot for a unnormated network with thus focus on angular data</i>
+To do this, lets look first at the loss of the autoencoder: Since it is basically just the difference between input and output<note simplifying chapter <ref losses> a bit>, it is a measurement about how good the autoencoder reproduces whatever we put into it<note this is also a bit of a simplification, since the normalization of the data matters a lot, but we will come back to this later>, and so by just calculating the loss on background data, we have a measure for the quality of the autoencoder. <ignore>Also, as argued for in the chapter about decicision quality <ref binclass>, we will use the AUC score to evalutate the classifier quality.</ignore> So basically we want to see a strong falling correlation between the loss of our network and the AUC score<note the lower the loss, the higher the AUC, since we train on QCD jets>, do we see this? We show here the loss in training against the decision quality of the corresponding network in this trainingstep. Please note that this is exactly what we want, since the correlation we want here, would mean, that in training, a classifier gets continously better<note yes you could argue, that it is enough when the highest AUC is reached at the lowest loss, but in practice this is not enough because the network does not always reach the same point><note it should be noted, that a network cancels the training when it does not improve for a certain number of epochs, so in theory you do not know if there may not even be a better classifier at a way later epoch, but with worse loss. This is usually assumed to be false, since overfitting defines the rest of training, but since we do not see basically any overfitting, this migth not be so easily ignored><note we should also note, that this is an analysis that we did only for a small fraction of all networks, since evalutating the quality of hunderts of networks takes considerable time, often even more than training itself. This is also why we change this method later, to consider only those epochs in which the network improves its loss>.
+<i wmode="True" f="none">(FORGOT THIS PLOT, AND CALCULATION TAKES TIME)A simple old AUC by epoch plot for a unnormalized network with thus focus on angular data</i>
 At first glance yes, almost all networks, at least since we consider them working autoencoder, are monotonously falling, but there some sidemarks: Most importantly is this one of those networks that we trained just to have a high AUC, and thus is a network that basically just compares angles to zero<note see chapter <ref secgae>>. This does not mean that this image is useless, as it shows, that at least a networks finds out that a way less useful feature should be ignored, but we should look at the AUC by epoch of a good network:
 <i wmode="True" f="none">(FORGOT THIS PLOT, AND CALCULATION TAKES TIME)a "good" abe</i>
 As you see, this relation is basically the same as before, except for two differences: The new networks are worse, and we do have way less epochs. This is since we stopped calculcating the AUC for each epoch, but now only calculcate it for each epoch in which the network improved<note this we did for computational reasons, not only to save time, but also disc space. And even though this makes our reasoning a bit less accurate, this should be a reasonable compromise>
 More interrestingly, please note a peculiarity in the preceding images: As you see the relations are almost linear. This is not neccesarily always the case, consider this image
 <i wmode="True" f="none">(FORGOT THIS PLOT, AND CALCULATION TAKES TIME)ABE nonorm like exp on trivial decompressor</i>
 It should be noted, that this image is still of the bad kind, focussing mostly on the angular part, but as you see, this image still shows a stricly falling relation, while this time it seams to be exponential in nature<note something like #c-exp(-d*x)#>. You could ask yourself what is better? The exponential one migth be limited in its quality, but is also easier to saturate. And this is most likely a feature of our network architecture, since you get this curve when you replace the decoder with a more trivial one. Interrestingly the quality in the linear seeming case cannot be linear, since the AUC has to always be below 1, so you could ask yourself how this curve continous. The obvious first assumption would be that both curves are expoential in nature, but we are not able to saturate the capacities of a nontrivially decoding network.
-As logically as this seems, testing this is a bit of a different story: Since apparently our networks dont reach the neccesary quality to saturise themself, this is hard, if not impossible to test. Which is why give up testing just one autoencoder, either by stopping to test just one network, or by testing other kind of classifiers, and as you will see, both suggest a similar, again falling quality curve for little losses: 
+As logically as this seems, testing this is a bit of a different story: Since apparently our networks don`t reach the neccesary quality to saturise themself, this is hard, if not impossible to test. Which is why give up testing just one autoencoder, either by stopping to test just one network, or by testing other kind of classifiers, and as you will see, both suggest a similar, again falling quality curve for little losses: 
 Lets start with multiple networks, instead of plotting parts of one network, we plot the result of multiple networks:
 <i wmode="rtvl_nl_log_2_4_5_6_7_8_9_1" f="none">(WILL BE REPLACED, ORANGE AND GREEN HAVE PARAM/GRAPH LIKE DECODER)ABE multi network with particle/graph like, and logged x axis</i>
-As you see, this relation is basically the same. Since the problem in this step was reproducability, you have a lot of different random network qualities, but, as already mentioned in chapter <ref secinv>, there is a strong  relation, that would be linear if the x axis would be linear, and that is growing, since it is trained on top data, and in our definitions, the optimal AUC would now be 0 instead of 1. The more interresting part is now those part at really small loss, that seams to deviate from the relation. These are two network types, that have a more complicated decoder, and as you see: They are definitely way better autoencoder, but are also less good classifier.<ignore> NOW TO LOOK AT ABE FOR THE BEST OF THOSE NETWORKS. ALSO WE CAN LOOK AT ONEOFF NETWORKS, since there quite similar to autoencoding, I would expect a similar relationship, but as you see here</ignore>
-Then consider chapter <ref secother>, and oneOff networks that show basically the same relation!
+As you see, this relation is basically the same. Since the problem in this step was reproducability, you have a lot of different random network qualities, but, as already mentioned in chapter <ref secinv>, there is a strong  relation, that would be linear if the x axis would be linear, and that is growing, since it is trained on top data, and in our definitions, the optimal AUC would now be 0 instead of 1. The more interresting part is now those part at really small loss, that seams to deviate from the relation. These are two network types, that have a more complicated decoder, and as you see: They are definitely way better autoencoder, but are also less good classifier.<ignore> NOW TO LOOK AT ABE FOR THE BEST OF THOSE NETWORKS. ALSO WE CAN LOOK AT oneoff NETWORKS, since there quite similar to autoencoding, I would expect a similar relationship, but as you see here</ignore>
+Then consider chapter <ref secother>, and oneoff networks that show basically the same relation!
 <i wmode="True" f="mabe3">AUC by epoch for oneoff showing a growing relation, that at some point starts to fall again</i>
-So lets us assume, the quality falls of again at some point, why could that be? One reason migth be, that signal and background are similar in certain features. Consider the following network: you feed it one particle only, but not only the 4 momentum, but also the mass: An autoencoder with the rigth compression size would learn to reconstruct the mass from the momentum 4 vector, probably more than it could ever find patterns in the 4 momentum itself. And if you consider that top quarks decay in a similar manner as QCD quarks, there are certainly similarities that an autoencoder should not focus on. This explains mostly why oneOff networks decrease in quality, since the just focus on one feature, but this is also a problem we can solve with a mixed approach, as shown in chapter <ref secmixed>. On the other hand, this migth explain why autoencoder this effect less, since they combine features, instead of relying on only one.
-That beeing said, this combination of features migth be the real problem: while talking about feature combination, we assumed that a saturated classifier, is an optimal classifier, but this is not actually the case. Consider the c addition explained in chapter <ref caddition> and tested in chapter <ref impro>: Any feature that is less useful has a bigger influence, that has to be compensated by a power 3 in this width, and since the leading pt particle is easier to reconstruct<note meaning the leading pt particle is less random> than for example the particle with the 7th highest pt, certain parts of the reconstruction are more or less useful, but the combination makes no difference between particles, so their combination will not be optimal, and thus the combination migth only be saturated with bad combination factors. In fact we can look at this, by looking at the partial networks from chapters <ref scale> and <ref scale2><note we should point out, that this is not entirely the same, since instead of adding particles, we here add bunches of 4 particles each, but we would not expect single particles to add differently than particle bunches, and training a graph on a single particle does not exactly make the best use of their relational setup>: when we dont add them together with their optimal factors, but with each factor beeing 1
+So lets us assume, the quality falls of again at some point, why could that be? One reason migth be, that signal and background are similar in certain features. Consider the following network: you feed it one particle only, but not only the 4 momentum, but also the mass: An autoencoder with the rigth compression size would learn to reconstruct the mass from the momentum 4 vector, probably more than it could ever find patterns in the 4 momentum itself. And if you consider that top quarks decay in a similar manner as QCD quarks, there are certainly similarities that an autoencoder should not focus on. This explains mostly why oneoff networks decrease in quality, since the just focus on one feature, but this is also a problem we can solve with a mixed approach, as shown in chapter <ref secmixed>. On the other hand, this migth explain why autoencoder this effect less, since they combine features, instead of relying on only one.
+That beeing said, this combination of features migth be the real problem: while talking about feature combination, we assumed that a saturated classifier, is an optimal classifier, but this is not actually the case. Consider the c addition explained in chapter <ref caddition> and tested in chapter <ref impro>: Any feature that is less useful has a bigger influence, that has to be compensated by a power 3 in this width, and since the leading pt particle is easier to reconstruct<note meaning the leading pt particle is less random> than for example the particle with the 7th highest pt, certain parts of the reconstruction are more or less useful, but the combination makes no difference between particles, so their combination will not be optimal, and thus the combination migth only be saturated with bad combination factors. In fact we can look at this, by looking at the partial networks from chapters <ref scale> and <ref scale2><note we should point out, that this is not entirely the same, since instead of adding particles, we here add bunches of 4 particles each, but we would not expect single particles to add differently than particle bunches, and training a graph on a single particle does not exactly make the best use of their relational setup>: when we don`t add them together with their optimal factors, but with each factor beeing 1
 <i f="splitscale" wmode="True">partial network addition, auc(gs) for factors=loss to power #-3# und factors=1</i>
 You see first, that the combination of equal factors reaches less good aucs, while still converging againt a certain AUC Value. You migth also notice that the equal factor one does only decrease at some point, but this happens, because as seen in chapter <ref caddition> there is a zone of factors, in which adding them together is not optimal, while still increasing the AUC score, so a decreasing classifier combination is just a more extreme version of nonoptimal combinations, that appears when two classifier are "too different". Also we just talked about different particles but same features, how about different features? How to you combine features with different meanings? C addition only works, since we assume that similar good reconstructed features have a similar width, and when you give this up, the width is no longer a measure only for quality but also for certain attributes of the feature itself<note consider the following network: it has some features and the same features times 4 as input (with some noise, that the autoencoder ignores), sure it will find out that all of its features are twice to reconstruct, and it will just learn to set the second batch of features to the first batch times 4, but how to combine those features now? If we just let the mse do this, the second batch is way more important, but this is unlogical, the best addition (ignoring noise for now) would be the first feature*4 + the second feature, and we can get it by simply normating the features in a certain way, but now consider the following: what if the noise of the second one has a different size than the noise of the first one? Then we would have to consider this, first multiply the first feature by 4 and then use c addition, but to do this, we need to know the multiplicative factor, and in general we wont>, and any algebraic combination becomes less useful, and also learnable approaches are tricky: Even though a perfect combination (a combination is a normalization here), migth result in the maximal AUC score, finding it can lead to nearly optimal maxima (as seen in <ref caddition>).Another idea migth be a learnable combination but here there is still the question of focus: When we talk about c addition, we also assume that there is a certain choice of things the network should focus on, and anything the network does not care to reproduce, is less accurate information <note there is also the inverse effect: when the network just copies a certain information, there is less decision quality in this feature potentially failing c addition>, but by making this focus basically learnable there is also a problem that you migth call normating and this is absolutely nontrivial. If you ignore normating your combination vector, the network will just learn zeros into it (a loss is always minimal if you compare zeros to zeros), but even when you assert that for example #Eq(abs(c),1)#, this does not neccesarily solve your problem, since the network migth focus entirely on the smallest feature, and even if you assert that the size of each feature is the same, it migth focus on the feature that is easiest to reconstruct, and thus it migth still be nonoptimal<note we think you migth be able to do solve this, by playing reconstruction complexity against size using c addition, but we have not tried this, and we expect it to be quite finicky>.
 As a final note in this subchapter: Please consider that oneoff networks should in theory solve all of this combination problems, since they work on minimizing their width only, and the optimal combination for a minimal width can be assumed to be the same as the combination for maximum AUC (see chapters <ref oomath> and <ref impro>), but as chapter <ref secmixed> and <ref secdata> show, they have their own problems.
@@ -35,7 +35,7 @@ As a final note in this subchapter: Please consider that oneoff networks should
 <subsubsection title="General problems" label="nogaegeneral">
 
 These were quite specific problems to top tagging and anomaly detection, but there are some more general problems working with autoencoders, we want to focus here on two:
-Firstly, it is not trivial to find the best compression size (see chapter <ref csize>. The lower this size, the more feature migth be compared to trivial values, and the higher it is, the more features migth be reconstructed perfectly and thus contain no classification information<note here the graph structure actually helps, since permutation invariance makes it nontrivial to get an true identity>. We choose my compressionsize by observing that a 4 node network is only invertible for a compression size of at least 9<note at least for 4 nodes constructed out of 3 features and flag, so a compression size of 9 means that i use #3/(4+flag)# of the inputsize as compressionsize>, but this still leaves the question open, how to increase the compression size with the number of nodes. You could argue that the more nodes there are, the more features are found that can be used to compress the input, but you could also argue that the added inputs are more random, and thus allow for less compression. This is why we just leave the fraction constant in the chapters that use bigger networks. You migth ask why we dont just test this again for a higher node count, but this would not only be quite timeintensive, but also only solve one more node size.
+Firstly, it is not trivial to find the best compression size (see chapter <ref csize>. The lower this size, the more feature migth be compared to trivial values, and the higher it is, the more features migth be reconstructed perfectly and thus contain no classification information<note here the graph structure actually helps, since permutation invariance makes it nontrivial to get an true identity>. We choose my compressionsize by observing that a 4 node network is only invertible for a compression size of at least 9<note at least for 4 nodes constructed out of 3 features and flag, so a compression size of 9 means that i use #3/(4+flag)# of the inputsize as compressionsize>, but this still leaves the question open, how to increase the compression size with the number of nodes. You could argue that the more nodes there are, the more features are found that can be used to compress the input, but you could also argue that the added inputs are more random, and thus allow for less compression. This is why we just leave the fraction constant in the chapters that use bigger networks. You migth ask why we don`t just test this again for a higher node count, but this would not only be quite timeintensive, but also only solve one more node size.
 and also there is this image (circle reconstruction problem compared to oneoffs, STILL WORKING ON)
 
 <ignore>
diff --git a/data/10anhang/05otheralgo b/data/10anhang/05otheralgo
index 6948f3683e812957ed24a79d071a979ac020e76c..734a9be581d7f1c8902df2f5726cbf87a03e818c 100644
--- a/data/10anhang/05otheralgo
+++ b/data/10anhang/05otheralgo
@@ -1,6 +1,6 @@
 <subsection title="Other algorithms" label="other">
 
-Since oneOff networks seem to have potential, that is just not used that well on jets, you could ask yourself if other classical methods work better. So this chapter serves as an introduction into several of those classical algorithms for finding signal events after training on background events, as well as a reasoning why this is not the case. The field these algorithms belong to, is called one class learning.
+Since oneoff networks seem to have potential, that is just not used that well on jets, you could ask yourself if other classical methods work better. So this chapter serves as an introduction into several of those classical algorithms for finding signal events after training on background events, as well as a reasoning why this is not the case. The field these algorithms belong to, is called one class learning.
 
 <subsubsection title="Support vector machines" label="whatssvm">
 Classicaly, SVMs are used to differentiate two sets of datapoints, by drawing lines between them, so that they are completely seperated. Instead of using deep learning, this problem could be solved analytically, even with the extension no longer requiring only lines, but a learnable transformation of a line, and thus allowing SVMs to not only work on linearly seperable data. This migth be more powerful, but still cannot handle every possible data distribution, and this problem stays the same for the one class learning version. Here you draw some shape (usually a circle) with some transformation (an ellipse) around your given background data, in a way that minimizes the volume. This restricted amount of possible shapes, can be useful, keeping the SVM from overfitting, but it also only allows it to learn certain distributions, so distributions like
@@ -8,7 +8,7 @@ Classicaly, SVMs are used to differentiate two sets of datapoints, by drawing li
 could not ever be learned. This is a problem, since a shape like this, could be the result of a simple rotational symmetry, which are not uncommon in physics, so as you migth expect, training an SVM<note implementent in sklearn> on QCD jets to find top jets does not result in any useful results (an AUC of #0.530#).
 
 <subsubsection title="k neirest neighbours" label="whatsnvm">
-Another usually quite useful algoritm is also an extension of an supervised task: Given two classes of vectors, you can classify each new point, by looking at the class of the vector that is closest to it, or at the mean of the classes of the #k# neirest vectors to it. This you can extend to the one class case, by setting the loss of one vector to be the mean of the differences to its #k# neirest neighbours. Since those known points are only background events, you can expect an abnormal event to have a higher loss, while background events are probably more similar to already known background events. The problem comes from its ability to overfit. This can easily be understood in the supervised case, since single weird background cases can lead to region in which no signal can be detected. You could say that autoencoder focus on the distribution of events, Support vector machines focus on the outliers of their distribution, while k neirest neighbour focusses on the whole volume, which means, it could solve the above distribution<refi oneclasscircle>, with an quite good AUC of #0.89# for #Eq(k,1)# and #0.96# for #Eq(k,100)#<note there is some overlay between signal and background in the image, which limits the AUC>, and in the jet case, this algorithm does better, reaching an AUC of ???, but it is still limited by the curse of dimensionality: One class learning algorithm usually work better on low dimensional inputs than on high dimensional one, and here this can be understood quite easily, since the volume of possible vectors grows exponentially with the dimension, while the number of trainingsamples wont chance to much, making the difference between each of the backgroundevents statistically bigger.
+Another usually quite useful algoritm is also an extension of an supervised task: Given two classes of vectors, you can classify each new point, by looking at the class of the vector that is closest to it, or at the mean of the classes of the #k# neirest vectors to it. This you can extend to the one class case, by setting the loss of one vector to be the mean of the differences to its #k# neirest neighbours. Since those known points are only background events, you can expect an abnormal event to have a higher loss, while background events are probably more similar to already known background events. The problem comes from its ability to overfit. This can easily be understood in the supervised case, since single weird background cases can lead to region in which no signal can be detected. You could say that autoencoder focus on the distribution of events, Support vector machines focus on the outliers of their distribution, while k neirest neighbour focusses on the whole volume, which means, it could solve the above distribution<refi oneclasscircle>, with an quite good AUC of #0.89# for #Eq(k,1)# and #0.96# for #Eq(k,100)#<note there is some overlay between signal and background in the image, which limits the AUC>, and in the jet case, this algorithm does better, reaching an AUC of ???, but it is still limited by the curse of dimensionality: One class learning algorithm usually work better on low dimensional inputs than on high dimensional one, and here this can be understood quite easily, since the volume of possible vectors grows exponentially with the dimension, while the number of trainingsamples wont change to much, making the difference between each of the backgroundevents statistically bigger.
 
 <subsubsection title="Isolation forests" label="whatsiforest">
 An isolation forest<note see <cite iforest>> works quite different to the algorithms explained before. Instead of defining what a normal event looks like, this algorithm tries to isolate anomalies. It does this, by randomly classifying points into a tree: Given some attributes, it picks a random one and a point at which to seperate the data by. By iterating this procedure, you build a tree, in which anormal data usually is seperated easier than the normal data, which means you can use the depth of a position in the tree as seperator. That beeing said, this algorithm migth be interresting, but still does not work very well, possible since it is also not immune to the curse of dimensionality, and so the result on QCD vs top is also only an AUC of 0.502.
diff --git a/data/10anhang/09bidentityproblems b/data/10anhang/09bidentityproblems
index f1c9af4579d10ffd32348e11c73a602b131723ea..b3edebd6d5e14345792e066c3bdd8b7f93561fa3 100644
--- a/data/10anhang/09bidentityproblems
+++ b/data/10anhang/09bidentityproblems
@@ -2,7 +2,7 @@
 
 An optimal autoencoder should be equivalent to the network with the compression size set to the input size. The problem here is, that this trivial model does not neccesarily reproduce its input perfectly. As described in chapter <ref gnn>, the graph update step is given by
 ##f(x_i*s_j+x_i*A_k**i*n_j**k)##
-and this kind of update step is not always invertible through another step. To see this, let us first ignore the activation function as #Eq(f(x),x)# and let us use a fixed size. Given 3 nodes of 2 features each, let the adjacency matrix and thus the graph be fixed to be<note please note that i set here the diagonal entries to zero, while in my implementation those are usually one, but this does not really matter, since this is just a chance in learnable parameters. Here this is done to simplify the following calculations>
+and this kind of update step is not always invertible through another step. To see this, let us first ignore the activation function as #Eq(f(x),x)# and let us use a fixed size. Given 3 nodes of 2 features each, let the adjacency matrix and thus the graph be fixed to be<note please note that i set here the diagonal entries to zero, while in my implementation those are usually one, but this does not really matter, since this is just a change in learnable parameters. Here this is done to simplify the following calculations>
 ##Matrix([[0,1,0],[1,0,1],[0,1,0]])##
 while the 2x2 matrices are general, we can use the kronecker product to convert them, corresponding to converting the 2 dimensional feature vector in an one dimensional one, into an corresponding matrix that can be multiplied to this new one dimensional feature vector. This matrix will then be given by
 ##Matrix([[s_00,s_01,n_00,n_01,0,0],
@@ -22,7 +22,7 @@ which can only be solved for
 but since #n# is given, the matrix cannot be invertible<note you could argue, that #n# is learnable, but this expects a bit much from the learning algorithm. More explicitely reducing the possible neigbourinteractions into a small subset reduces the possibilities of the learning algorithm and the worth of the graph drastically>.
 
 You could ask yourself if this is actually a problem, since even though two nonactivated update steps cannot invert themself, but surely a bunch of update steps are invertible together. We also assumed the adjacency matrix to be the same, which does not actually has to be the case. And even if not, since the compression size is not the same as the input size, the problem is anyway different.
-Sadly this is not something that we are able to easily calculate, but what we can do, is test this experimentally. As shown above, any graph update step can be rewritten as an product with a specific matrix. This allows us to create an inverse update step, that is equivalent to the normal one, except for a numerical inverse of the update matrix and train networks using those inverse update steps to decompress our data (You could ask yourself if the update matrix is invertible, and in general it is not (a trivial example migth be no neighbour interaction and a nonivertible self interaction<note aka a no self interaction>), but in practice this is a problem that can be controlled: It happens that the function used (tf.linalg.inv) fails, but this is rare, can be controlled by the initialiser of those matrices, and even if it fails, the documentation states that this function migth<note yes, the documentation is not very precise what happens in such a case> just return noise instead of showing errors. And considering that having a matrix that is not invertible requires each parameter to be exactly tuned, this can be ignored by the parameters constantly chancing. A bigger problem, and the reason, why we do not use these invertible matrices in each decompression phase are those matrices that are nearly uninvertible (have a determinant very close to zero). Since the determinants of the inverts of those matrices are huge, they can amplify noise and thus confuse the minimizationalgorithm. In practice that means that a network that once has reached a quite low loss, can have quite a bigger loss after a couple more training steps). These networks are now tested in two situations, first compared to a normal , quite good, Network, which results in a less smooth learning curve and/but (I SADLY FORGOT THOSE NETWORKS, SO THIS RESULS APPEAR A BIT LATER). Later we test those networks for the identity case. This means not only a compression size of the same size as the input size <note and also no compression steps. This migth seem trivial, but is actually not, since you could interpret a compression as chanching the distribution of features over the number of particles and the number of features for each particle>, but also afterwards, networks that compress, but are initialised to do nothing (I SADLY FORGOT THOSE NETWORKS, SO THIS RESULS APPEAR A BIT LATER).
+Sadly this is not something that we are able to easily calculate, but what we can do, is test this experimentally. As shown above, any graph update step can be rewritten as an product with a specific matrix. This allows us to create an inverse update step, that is equivalent to the normal one, except for a numerical inverse of the update matrix and train networks using those inverse update steps to decompress our data (You could ask yourself if the update matrix is invertible, and in general it is not (a trivial example migth be no neighbour interaction and a nonivertible self interaction<note aka a no self interaction>), but in practice this is a problem that can be controlled: It happens that the function used (tf.linalg.inv) fails, but this is rare, can be controlled by the initialiser of those matrices, and even if it fails, the documentation states that this function migth<note yes, the documentation is not very precise what happens in such a case> just return noise instead of showing errors. And considering that having a matrix that is not invertible requires each parameter to be exactly tuned, this can be ignored by the parameters constantly changing. A bigger problem, and the reason, why we do not use these invertible matrices in each decompression phase are those matrices that are nearly uninvertible (have a determinant very close to zero). Since the determinants of the inverts of those matrices are huge, they can amplify noise and thus confuse the minimizationalgorithm. In practice that means that a network that once has reached a quite low loss, can have quite a bigger loss after a couple more training steps). These networks are now tested in two situations, first compared to a normal , quite good, Network, which results in a less smooth learning curve and/but (I SADLY FORGOT THOSE NETWORKS, SO THIS RESULS APPEAR A BIT LATER). Later we test those networks for the identity case. This means not only a compression size of the same size as the input size <note and also no compression steps. This migth seem trivial, but is actually not, since you could interpret a compression as chanching the distribution of features over the number of particles and the number of features for each particle>, but also afterwards, networks that compress, but are initialised to do nothing (I SADLY FORGOT THOSE NETWORKS, SO THIS RESULS APPEAR A BIT LATER).
 
 So finally we see those invertibility problems as another kind of loss. Next to allowing only for #n# out of #m# features to be used, these #n# features have to work around the structure of the graph. This means, that comparable autoencoder of a non graph type work better for smaller compression size. This migth seem terrible, but we think it only means, that each compression size for a graph network is equivalent to a smaller one for a nongraph network, and thus, some compression sizes close to the maximum are impossible. Finally, you could even see this as an benefit for graph autoencoder, since choosing the rigth compression size is not a trivial task (see chapter <ref csize>), and this gives you kind of a regulator, saving you from choosing a too high one. Also please note, that the ability for the autoencoder to reconstruct data, does not imply anything concerning its effectiveness as classifier<ignore><note and even concerning chapter (ENTER chapter), the compressed dataspace is not effected (Or even regulated, since simply sopiying is no longer easily possible)></ignore>
 
diff --git a/data/10anhang/15mnist.py b/data/10anhang/15mnist.py
index 61b2ddee4011cea564adf00fddb12e962f3fe5ef..07ca47c375b71535c49d2592369a0a5726965f74 100644
--- a/data/10anhang/15mnist.py
+++ b/data/10anhang/15mnist.py
@@ -1,2 +1,2 @@
-<subsection title="Using oneOff networks on MNist data" label="amnist">
+<subsection title="Using oneoff networks on MNist data" label="amnist">
 
diff --git a/data/10anhang/20metrikana b/data/10anhang/20metrikana
index 155fd6fbdbd9d5fb87051aa39be3cefeb68ca12f..3fc7263b3c3ab2fecd5cb6debfee45d741e90e53 100644
--- a/data/10anhang/20metrikana
+++ b/data/10anhang/20metrikana
@@ -4,7 +4,7 @@ As explained in chapter <ref imgsetup>, our topK algorithm, on which all graphes
 <i f="none" wmode="True">metrik of usual networks</i>
 (ENTER understanding)
 
-Another thing, that you can do with this, is chance the input definitions. One obvious choice would be to use a simple 4 vector. This results in the following metrik
+Another thing, that you can do with this, is change the input definitions. One obvious choice would be to use a simple 4 vector. This results in the following metrik
 <i f="none" wmode="True">metrik of 4vector networks</i>
 Here you see two interesting things: firstof, the first value, corresponding to the jet energy, is negative. In the current implementation, this means, that two points, are more likely connected, the more different their energy is. You could interpret this as unphysical and try to justify using this, that this choice of input parameters is not a good one<note you can much more easily justify this, since the resulting training is way less stable>, but you could also note, that this difference is essentially a minkowski metrik. That beeing said, we are not sure, if this has any significance, as compared to the other monkowski metrik that appears in this thesis (chapter <ref oometrik>), this cannot simply be interpreted as a mass, since
 ##(p_1-p_2)**2-(E_1-E_2)**2##
diff --git a/data/10anhang/30topK b/data/10anhang/30topK
index e7a674a7e4427cd2729df5b407893e9908a55137..7209787a9b42bf477e213b582c19b8ecaa6207eb 100644
--- a/data/10anhang/30topK
+++ b/data/10anhang/30topK
@@ -6,7 +6,7 @@ Writing a topK layer to connect each node to its #K# neighbours is actually not
 <i f="none" wmode="True">example of impos graph (3, 2 of them same distance)</i>
 when two nodes are of the same distance, which to connect? We simply connect both, as states in which both are of the exactly same distance are very rare, but there is a more complicated problem here:
 <i f="none" wmode="True">example of impos graph (3 2 of them together)</i>
-how to connect to nodes that dont have neighbours open? we solve this by no longer requiring the adjacency matrix to be symmetric, and thus the graph to be directed, but you could think about chancing this, by symmetrizing the matrix, either in a way that requires both or either direction to be connnected
+how to connect to nodes that don`t have neighbours open? we solve this by no longer requiring the adjacency matrix to be symmetric, and thus the graph to be directed, but you could think about changing this, by symmetrizing the matrix, either in a way that requires both or either direction to be connnected
 
 (ENTER RESULTS)
 
@@ -14,7 +14,7 @@ how to connect to nodes that dont have neighbours open? we solve this by no long
 <subsubsection title="How topK migth actually not be the best idea" label="atopkwhy">
 
 
-Since this is just an appendix, I want to take the time, of starting this chapter with a story: A while ago, I was in a chapel. This does not happen very often, and judging from the chapel, it is also not used very often. One thing you immendiatly notice, is that this chapel did not have any banks, but used a lot of chairs, and since at that moment, I desperatly wanted to think about something else, I became faszinated by those chairs: For some reason, even though these chairs are fairly unordered, we dont think of them as just #n# chairs, but as #r# rows of #c# chairs each. Why? how do we abstract those unordered amount of chairs into need rows and collumns, and do we have an algorithm that can do this for us? Since this is still a thesis about graph autoencoder and not my journal, you migth guess the relation: if you simply draw graph connection between each neighbouring chair, you generate a graph like this
+Since this is just an appendix, I want to take the time, of starting this chapter with a story: A while ago, I was in a chapel. This does not happen very often, and judging from the chapel, it is also not used very often. One thing you immendiatly notice, is that this chapel did not have any banks, but used a lot of chairs, and since at that moment, I desperatly wanted to think about something else, I became faszinated by those chairs: For some reason, even though these chairs are fairly unordered, we don`t think of them as just #n# chairs, but as #r# rows of #c# chairs each. Why? how do we abstract those unordered amount of chairs into need rows and collumns, and do we have an algorithm that can do this for us? Since this is still a thesis about graph autoencoder and not my journal, you migth guess the relation: if you simply draw graph connection between each neighbouring chair, you generate a graph like this
 <i f="none" wmode="True">chair graphs</i>
 This graph is easily abstracted and reconstructed using the algorithms explained in chapter <ref thirdworking>: You can use one axis of the diagram (probably y) as seperation variable. This results in subgraphs for each row, which get compressed into one object that we migth call row. Then, this rowgraph tensorproducted to the sittingplan as graph of rows gives to original graph. Yes, there are problems, notably you need to know the number of rows before<note as the compression number has to be a factor of the number of chairs, and finding half rows or double rows, still would be good representation, this is actually not that big of an problem>, but this is still an application of a point cloud, that is better compressed using my graph autoencoder, than every other encoder. There is just one problem: the graph we use is not a topK graph!
 To be precise: A topK graph is defined by each node having the same number of neighbours, but the chairs next to edge have less neighbours than chairs in the middle of the chapel. This means that this graph cannot be the output of the topK algorithm. In fact you can show quite easily, that only the tensorproduct of a topK graph with a topK graph returns a topK graph. 
diff --git a/data/10anhang/32oomath b/data/10anhang/32oomath
index 7bf479e6e26161f97ea59f4302daac9ef45906bb..117fb55577ba236542895d0fdd1ace50bc3e04e4 100644
--- a/data/10anhang/32oomath
+++ b/data/10anhang/32oomath
@@ -1,12 +1,12 @@
 <subsubsection title="oneoff math" label="oomath">
 Before we talk about how this works, lets talk about the math behind this idea a bit, especially how this kind of networks should handle multiple kinds of information. To do this, let us consider a simple model: Each feature is build out of two gaussian distributions, the first distribution describes the training/background data, and thus has a mean of 1 and some width #sigma_1# and the second one describes the signal data, it has a mean of #mu# and a width of #sigma_2#. This means the decision quality of this feature can be described by #Eq(s,abs(mu-1)/sqrt(sigma_1**2+sigma_2**2))#. The higher #s# is, the bigger the difference between both peaks, and the better the seperation and thus the higher the auc. This migth remind the reader of the math considered in the chapter about c addition (<ref caddition>), and it actually concludes, that by considering how to combine two background, the math is exactly same as for c addition:
-Given two distributions, with width #s_1# and #s_2# and mean values of #1#, you can combine both distributions into one distribution with smaller width. This width of the distibution #(d_1+c*d_2)/(1+c)# is given by #sqrt(s_1**2+c**2*s_2**2)/(1+c)# while the mean is still #1#. This function is exactly the same, as was minimized to find the combination with the best AUC in chapter <ref caddition>. This migth suggest, that the resulting combination of an oneOff network is the combination with the highest possible AUC. Sadly this is simply not sp easy: The problem are the assumptions made in the chapter about c addition(<ref caddition>): We set the distance between the background and the signal peak to be constant, which results in the width of the distributions to be the only important thing to consider, when combining two distributions. This is fine, when considering features of similar kind, since you can assume their distributions to be similar, but does not anymore here: And when you assume this distance to be more or less random, the calculation becomes a bit more complicated. In fact you can assume, that any other possible peak could be some sort of signal data, that you want to exclude (see chapter <ref impro>).
-So consider the following model: Given a background peak around 1 width a certain width #s_1#, and an improvement of this peak, being more focussed with a width #LessThan(s_2,s_1)#: Which peak is more probable to seperate a random signal from the background? Since the second peak is less wide, it is less probable for a signal peak to overlap it and thus probably results in a higher AUC score<note> To be more precise, you optimize the background peak by improving on the function that generates it, this clearly also changes the signal peak, but does this in a more or less random manner. It could improve the AUC, but could also hurt it. The point is, that this chance is random, and thus on average, optimizing an oneOff network migth be useful. We do this more precisely in chapter <ref impro></note>
+Given two distributions, with width #s_1# and #s_2# and mean values of #1#, you can combine both distributions into one distribution with smaller width. This width of the distibution #(d_1+c*d_2)/(1+c)# is given by #sqrt(s_1**2+c**2*s_2**2)/(1+c)# while the mean is still #1#. This function is exactly the same, as was minimized to find the combination with the best AUC in chapter <ref caddition>. This migth suggest, that the resulting combination of an oneoff network is the combination with the highest possible AUC. Sadly this is simply not sp easy: The problem are the assumptions made in the chapter about c addition(<ref caddition>): We set the distance between the background and the signal peak to be constant, which results in the width of the distributions to be the only important thing to consider, when combining two distributions. This is fine, when considering features of similar kind, since you can assume their distributions to be similar, but does not anymore here: And when you assume this distance to be more or less random, the calculation becomes a bit more complicated. In fact you can assume, that any other possible peak could be some sort of signal data, that you want to exclude (see chapter <ref impro>).
+So consider the following model: Given a background peak around 1 width a certain width #s_1#, and an improvement of this peak, being more focussed with a width #LessThan(s_2,s_1)#: Which peak is more probable to seperate a random signal from the background? Since the second peak is less wide, it is less probable for a signal peak to overlap it and thus probably results in a higher AUC score<note> To be more precise, you optimize the background peak by improving on the function that generates it, this clearly also changes the signal peak, but does this in a more or less random manner. It could improve the AUC, but could also hurt it. The point is, that this change is random, and thus on average, optimizing an oneoff network migth be useful. We do this more precisely in chapter <ref impro></note>
 <i f="img_doublepeak" wmode="True">(c4/16) Two different widths of a background peak, resulting in different overlapping to the signal peak</i>
-This combination of two improving methods, C addition for similar features, and statistical improvement for asimilar features, is why we think, that oneOff networks migth work well, but there are two caveats we need to talk about here: First, this optimization does not help at all, when the distance between the background the signal peak is zero, since when the oneOff network focusses on something that is the same in background and signal, making a distribution less wide, only results in both getting smaller. In practice this becomes only a big problem, when you have trivial inputs, consider the case of the autoencoder discussed at the beginning of the chapter: If you chance the normalization to have a trivial #1# in on of the other outputs, you lose all decision power of the flag variables. A possible solution to this problem will be discussed in the next chapter <ref secmixed>.
-Another caveat that should be mentioned, is the loss you use for training an oneOff network. The first choice migth be to just minimise #(x-1)**2#, but if you try this, you notice that the resulting mean is not #1# but smaller than #1#. To understand this, consider the following model: given a gaussian peak #I# with a width #sigma# and a mean of #1#, which distribution #mu*I#, with a constant #mu#, minimizes #(mu*I-1)**2#? This loss can be written<note>when approximating the number of training samples as infinite</note> as
+This combination of two improving methods, C addition for similar features, and statistical improvement for asimilar features, is why we think, that oneoff networks migth work well, but there are two caveats we need to talk about here: First, this optimization does not help at all, when the distance between the background the signal peak is zero, since when the oneoff network focusses on something that is the same in background and signal, making a distribution less wide, only results in both getting smaller. In practice this becomes only a big problem, when you have trivial inputs, consider the case of the autoencoder discussed at the beginning of the chapter: If you change the normalization to have a trivial #1# in on of the other outputs, you lose all decision power of the flag variables. A possible solution to this problem will be discussed in the next chapter <ref secmixed>.
+Another caveat that should be mentioned, is the loss you use for training an oneoff network. The first choice migth be to just minimise #(x-1)**2#, but if you try this, you notice that the resulting mean is not #1# but smaller than #1#. To understand this, consider the following model: given a gaussian peak #I# with a width #sigma# and a mean of #1#, which distribution #mu*I#, with a constant #mu#, minimizes #(mu*I-1)**2#? This loss can be written<note>when approximating the number of training samples as infinite</note> as
 ##gauss((x-1)**2,1-mu,mu*sigma)##
 where we write #gauss(a,mu,sigma)# as the application of an gaussian kernel with mean #mu# and width #sigma# around #a#.
-This function has the same minima as<note>Since the constant part does not chance the minima, and the linear part is zero by symmetry</note>
+This function has the same minima as<note>Since the constant part does not change the minima, and the linear part is zero by symmetry</note>
 ##Eq(gauss(x**2,1-mu,mu*sigma),(1-mu)**2+mu**2*s**2)##
 This function is minimal at #Eq(mu,1/(1+s**2))#. Since this value is not always #1#, training on this loss, while comparing the signal peak to 1, does not work. You can fix this, by simply training on a different loss #(mean(x)-1)**2+std(x)**2#<note>where #mean(x)# returns the mean of the input distribution, while #std(x)**2# returns its variance</note> works quite well, or you can just read just the mean of the output distribution, by subtracting the mean of the background peak, instead of 1. Both methods work with similarly good results, even though their output is not always the same. In the following basically always readjusted means are used.
\ No newline at end of file
diff --git a/data/10anhang/33combinatorics b/data/10anhang/33combinatorics
index 673e53e07163d72eafc23e326ca07a6d18a00439..3f88767fcff5570014f45ce12fd21012fa8655fd 100644
--- a/data/10anhang/33combinatorics
+++ b/data/10anhang/33combinatorics
@@ -1,9 +1,9 @@
 <subsection title="Self improving oneoff networks" label="impro">
 
-We justified oneOff networks before (see <ref oomath>) by showing that the factor needed to combine two double gaussian features is the same, when we talk about minimizing the the width of the first peak, or when we use c addition to find the maximum auc value. That migth seem great, but there is one assumption, that we kind of glossed over a bit in the chapter about c addition: The difference between both peaks, is not neccesarily defined by the size of the first peak. So here we want to go into detail about why this is not neccesarily a problem, and what consequences can follow from breaking this assumtion.
+We justified oneoff networks before (see <ref oomath>) by showing that the factor needed to combine two double gaussian features is the same, when we talk about minimizing the the width of the first peak, or when we use c addition to find the maximum auc value. That migth seem great, but there is one assumption, that we kind of glossed over a bit in the chapter about c addition: The difference between both peaks, is not neccesarily defined by the size of the first peak. So here we want to go into detail about why this is not neccesarily a problem, and what consequences can follow from breaking this assumtion.
 The way we try to understand this, is by looking at width vs auc plots, of different c values combining random gaussian double peaks, that have each a random width between #0# and #2#, while having fixed means of #0# and #1# each<note this is still general, since translation and scale invariance give us 2 degrees of freedom per doublepeak>. After simulating a lot of random distributions, three nontrivial<note with trivial we mean relations that are defined by at least one doublepeak with an AUC of 1> classes seem to emerge
 <i f="oosym18" f2="oosym12" f3="oosym13" wmode="True">(test/oosym/imgs/spec)the three kinds of simulated AUC by loss plots</i>
-As you see, the first relation is pretty much perfect: the lower the width of the first combined resulting peak, the higher the AUC value is. This is the class we want, and we would get if the assumption would be true. Sadly  this is not the only possible result and the second class is not that optimal: These are distributions of supoptimal combination, where the lowest loss, does not result in the highest AUC, but at least into some value that is close to the expected optimum. This class appears in different levels of accuracy, reaching from distributions, that are nearly indifferable from the optimal case, to some, that are definitely not good. Finally the third class, contains combinations that are completely suboptimal: The optimal AUC value is reached at a basically terrible loss, and by decreasing the loss, the AUC becomes bad again. These are what you migth call traps: a very bad classifier is hidden behind a small initial distribution. You can easily see why this cannot ever be filtered out, by considering the case of a trivial feature, that is just always (for signal and background) #1#: the oneoff network will focus entirely on it, since it can reach a loss that is exactly zero, ignoring every feature that would be better at classification, and thus reach a useless classification score, and by looking only at the background distribution, there is nothing you can do<note except for using a different algorithm, for example a SVM>. Please note, that in this case, the autoencoder would actually solve the problem: since the feature is trivial, it will be filtered out, and thus cannot be learned from the oneOff network. Combine this with the fact that, from a quick simulation, this case does not seem to appear to commonly
+As you see, the first relation is pretty much perfect: the lower the width of the first combined resulting peak, the higher the AUC value is. This is the class we want, and we would get if the assumption would be true. Sadly  this is not the only possible result and the second class is not that optimal: These are distributions of supoptimal combination, where the lowest loss, does not result in the highest AUC, but at least into some value that is close to the expected optimum. This class appears in different levels of accuracy, reaching from distributions, that are nearly indifferable from the optimal case, to some, that are definitely not good. Finally the third class, contains combinations that are completely suboptimal: The optimal AUC value is reached at a basically terrible loss, and by decreasing the loss, the AUC becomes bad again. These are what you migth call traps: a very bad classifier is hidden behind a small initial distribution. You can easily see why this cannot ever be filtered out, by considering the case of a trivial feature, that is just always (for signal and background) #1#: the oneoff network will focus entirely on it, since it can reach a loss that is exactly zero, ignoring every feature that would be better at classification, and thus reach a useless classification score, and by looking only at the background distribution, there is nothing you can do<note except for using a different algorithm, for example a SVM>. Please note, that in this case, the autoencoder would actually solve the problem: since the feature is trivial, it will be filtered out, and thus cannot be learned from the oneoff network. Combine this with the fact that, from a quick simulation, this case does not seem to appear to commonly
 <table caption="Distribution of types of loss vs AUC plots for random gaussian double peaks" label="lossdist" c="2" modus="full">
 <hline>
 <tline "Class~Number">
@@ -21,8 +21,8 @@ As you see, the first relation is pretty much perfect: the lower the width of th
 
 and you can suggest that this will not be a problem.
 
-But to test this, we have to work on actual data, so this is the loss vs AUC relation for a network trained on the compressed space of top jets, trying to find QCD as signal. Please note, that there is a huge difference to the simple case of optimally adding gaussian double peaks: Firstly, there is an unknown<note unknown, as we cannot really find out, how many informations the network uses to classify a feature. Since l2 normalised networks usually dont set trained parameters to zero, and any gaussian peak can usually help reduce the width of a peak (Central limit theorem), it is actually reasonable to assume, that the whole compressed space, as a transformation of a here 9 dimensional feature vector, is used> number of features beeing combined. This wont chance to much, since the quadratic addition of #n# features can be understood as the quadratic addition of one feature with the quadratic addition of #n-1# features, but migth be valuable to keep in mind. Secondly, instead of looking at different values of a #c# factor, combining the peaks, we only look those c values, the network considers at the end of each epoch: That means, we could overlook an optimal AUC in the middle of an epoch, or even miss a good classifier never considered by the network.
+But to test this, we have to work on actual data, so this is the loss vs AUC relation for a network trained on the compressed space of top jets, trying to find QCD as signal. Please note, that there is a huge difference to the simple case of optimally adding gaussian double peaks: Firstly, there is an unknown<note unknown, as we cannot really find out, how many informations the network uses to classify a feature. Since l2 normalised networks usually don`t set trained parameters to zero, and any gaussian peak can usually help reduce the width of a peak (Central limit theorem), it is actually reasonable to assume, that the whole compressed space, as a transformation of a here 9 dimensional feature vector, is used> number of features beeing combined. This wont change to much, since the quadratic addition of #n# features can be understood as the quadratic addition of one feature with the quadratic addition of #n-1# features, but migth be valuable to keep in mind. Secondly, instead of looking at different values of a #c# factor, combining the peaks, we only look those c values, the network considers at the end of each epoch: That means, we could overlook an optimal AUC in the middle of an epoch, or even miss a good classifier never considered by the network.
 <i f="abeoo" wmode="True">(1095 abeoo)</i>
-As you see, the AUC seems to fall with the width of the first distribution(this is what we want, since we train on top jets), but not reaching the true optimal value. Comparing this to the theoretical expectation is not that easy: the most obvious reason why this does not matter migth be the inaccuracy of the AUC values: the value migth not be optimal, but is very close to the optimal value, while the AUC seems to fluctuate about more than the expected value. More interresting is the relation at less than optimal AUC values. Sadly there are not that many points here. This correlates to the fact, that oneOff networks gain most of their progress in their first epochs(if not epoch), but the points that we see, seem to fit quite nicely to one, if not two lines, building the tails of the theoretically expected relations for an at least close to optimal case. That beeing said, even ignoring the theoretical expectation, and the possibility of another even better combination, this is a relation that validates our training procedure, suggests that the initial autoencoder works at removing traps, and migth even suggest, that one of the reasons, multiple oneOff networks combined are better than only one, comes from the fact, that multiple distributions reduce the noise in the AUC loss relation, and thus gain statistically better AUC values<note even though it should be noted, that this cannot be the only effect, since this combination usually results in about an increase of 5\%, while this only seems to account for at most 1\%>.
+As you see, the AUC seems to fall with the width of the first distribution(this is what we want, since we train on top jets), but not reaching the true optimal value. Comparing this to the theoretical expectation is not that easy: the most obvious reason why this does not matter migth be the inaccuracy of the AUC values: the value migth not be optimal, but is very close to the optimal value, while the AUC seems to fluctuate about more than the expected value. More interresting is the relation at less than optimal AUC values. Sadly there are not that many points here. This correlates to the fact, that oneoff networks gain most of their progress in their first epochs(if not epoch), but the points that we see, seem to fit quite nicely to one, if not two lines, building the tails of the theoretically expected relations for an at least close to optimal case. That beeing said, even ignoring the theoretical expectation, and the possibility of another even better combination, this is a relation that validates our training procedure, suggests that the initial autoencoder works at removing traps, and migth even suggest, that one of the reasons, multiple oneoff networks combined are better than only one, comes from the fact, that multiple distributions reduce the noise in the AUC loss relation, and thus gain statistically better AUC values<note even though it should be noted, that this cannot be the only effect, since this combination usually results in about an increase of 5\%, while this only seems to account for at most 1\%>.
 
 
diff --git a/data/10anhang/34oonophys b/data/10anhang/34oonophys
index 664b541877e8163d3993a6f69fa013c1f49de5da..f0407f416212d0df30e68fd94615383aa565733c 100644
--- a/data/10anhang/34oonophys
+++ b/data/10anhang/34oonophys
@@ -1,4 +1,4 @@
 
 <subsubsection title="oneoff outside of physics" label="oomnist">
-This apparently unused potential led us to try them out on more classical evaluation datasets, and we found a paper(ENTER REFERENCE), that not only works fairly similar to oneOff networks<note>They use something called a support vector machine, which is probably most easily described as an algorithm that draws a circle like shape around the known datapoints, and classifies everything inside of the shape to be background, and everything outside to be signal. There main idea is to make the shape to be learnable in a deep way. So the main difference to oneoff networks is the fact that here there is a certain region, with the smallest possible size, optimal to be in for the background events, while in oneoff networks the only values that are optimal are exactly one</note>, but also evaluates them quite thourougly on MNIST<note>MNIST is a set of handwritten digits, that is often used to test new algoritms <cite mnist></note>. One algorithm they test their algorithm against is based on autoencoders while another uses GANs, and they constantly outperform them. We test here oneOffs on the following task: Given drawings of the number 7, how well can you detect other numbers. They provide also the results from assuming every other number to be the background, and we provide our results in the appendix <ref amnist>, but here we focus on 7 for now, since it seems not to easy, while also not beeing to hard of a task. They reach an AUC score of #0.946# with an error of #0.009#, while oneOffs reach a quality of #0.914# with and error of #0.018#. You could see this, and think that again, they have potential, but they are definitely worse than the reference paper, but this would ignore one fact: There approach does take the whole datavector to retrieve its loss, while our only takes some part, and by retraining the oneoff network, they do not predict the exact same thing<note>on average there is a correlation of about #0.6# between each retraining</note>. This means that it should be easy to combine multiple runs into one good classifier, and the math for this (see chapter <ref caddition>) is even easier here, since every network could reach the same quality, you can set #Eq(c,1)# and just add each value of #abs(x-1)# together. If you do this with enough reruns<note>We used here 25 runs</note>, the AUC converges against a value of #0.981#, beating the comparison paper, and thus showing the true potential of oneoff networks for one class learning!
+This apparently unused potential led us to try them out on more classical evaluation datasets, and we found a paper(ENTER REFERENCE), that not only works fairly similar to oneoff networks<note>They use something called a support vector machine, which is probably most easily described as an algorithm that draws a circle like shape around the known datapoints, and classifies everything inside of the shape to be background, and everything outside to be signal. There main idea is to make the shape to be learnable in a deep way. So the main difference to oneoff networks is the fact that here there is a certain region, with the smallest possible size, optimal to be in for the background events, while in oneoff networks the only values that are optimal are exactly one</note>, but also evaluates them quite thourougly on MNIST<note>MNIST is a set of handwritten digits, that is often used to test new algoritms <cite mnist></note>. One algorithm they test their algorithm against is based on autoencoders while another uses GANs, and they constantly outperform them. We test here oneoffs on the following task: Given drawings of the number 7, how well can you detect other numbers. They provide also the results from assuming every other number to be the background, and we provide our results in the appendix <ref amnist>, but here we focus on 7 for now, since it seems not to easy, while also not beeing to hard of a task. They reach an AUC score of #0.946# with an error of #0.009#, while oneoffs reach a quality of #0.914# with and error of #0.018#. You could see this, and think that again, they have potential, but they are definitely worse than the reference paper, but this would ignore one fact: There approach does take the whole datavector to retrieve its loss, while our only takes some part, and by retraining the oneoff network, they do not predict the exact same thing<note>on average there is a correlation of about #0.6# between each retraining</note>. This means that it should be easy to combine multiple runs into one good classifier, and the math for this (see chapter <ref caddition>) is even easier here, since every network could reach the same quality, you can set #Eq(c,1)# and just add each value of #abs(x-1)# together. If you do this with enough reruns<note>We used here 25 runs</note>, the AUC converges against a value of #0.981#, beating the comparison paper, and thus showing the true potential of oneoff networks for one class learning!
 <i f="examples" wmode="True">(752/draw) On the top: The 5 least 7 like 7th in the training set. On the bottom: The 5 most 7 like not 7th in the evaluation sample</i>
\ No newline at end of file
diff --git a/data/10anhang/35oometrik b/data/10anhang/35oometrik
index 8c3f85e83558042aa1565141dcb04acf59779937..99b3be8a91fd7ca7bbfb3ea4575ab958d0aeb6ca 100644
--- a/data/10anhang/35oometrik
+++ b/data/10anhang/35oometrik
@@ -1,6 +1,6 @@
-<subsubsection title="Some physical interpretability for oneOff networks" label="oometrik">
-Another example, why oneOff networks migth be quite useful, comes from our experiments to understand them more <ignore>(see chapter (ENTER chapter))</ignore>. Instead of constructing arbitrary features by utilising deep networks, the algorithm used here only combines input features in a linear way. The data we work on here is provided by cern open data as two lepton events from the 2010 datasets. Momentum 4 vectors of muons<cite setmuon> as background and of electrons<cite setelectron> as signals. These 4 vectors are multiplied with a linear metrik, reducing it into one dimension, that is evaluated to minimize #(abs(p**(mu)*g_mu_nu*p**(nu))-1)**2#. This results in the network learning the following metrik
-<table caption="Learned metrik values of oneOff networks trained on muon events" label="oneoffmuon" c="5" mode="classic">
+<subsubsection title="Some physical interpretability for oneoff networks" label="oometrik">
+Another example, why oneoff networks migth be quite useful, comes from our experiments to understand them more <ignore>(see chapter (ENTER chapter))</ignore>. Instead of constructing arbitrary features by utilising deep networks, the algorithm used here only combines input features in a linear way. The data we work on here is provided by cern open data as two lepton events from the 2010 datasets. Momentum 4 vectors of muons<cite setmuon> as background and of electrons<cite setelectron> as signals. These 4 vectors are multiplied with a linear metrik, reducing it into one dimension, that is evaluated to minimize #(abs(p**(mu)*g_mu_nu*p**(nu))-1)**2#. This results in the network learning the following metrik
+<table caption="Learned metrik values of oneoff networks trained on muon events" label="oneoffmuon" c="5" mode="classic">
 <hline>
 <tline " ~#E#~#p_1#~#p_2#~#p_3#">
 <hline>
@@ -13,7 +13,7 @@ Another example, why oneOff networks migth be quite useful, comes from our exper
 </table>
 
 As you see, the result is very similar to a minkowski metrik: The nondiagonal parts are zero in the range of numerical uncertainity (and symmetric for 5 digits behind the commata), the signs are randomly this way, because of the absolut value in the loss function and the absolute value of the diagonal parts scales the resulting expected output of #1# that the loss expects. Other than this, this simple network is able to understand itself, that characterising a particle is best done through what we call its mass. That beeing said, the AUC score is not optimal, only reaching #0.5988#, even though we can improve this, by assuming the metrik to be strictly diagonal, which results in a learned metrik of
-<table caption="Learned metrik values for a diagonal metrik oneOff networks trained on muon events" label="oneoffmuondiag" c="4" mode="free">
+<table caption="Learned metrik values for a diagonal metrik oneoff networks trained on muon events" label="oneoffmuondiag" c="4" mode="free">
 <hline>
 <tline "#E#~#p_1#~#p_2#~#p_3#">
 <hline>
diff --git a/data/10anhang/36whyoneoffsfail b/data/10anhang/36whyoneoffsfail
index 570be51c10b4dbf348b53243356671e9209fea83..004ffb36d1e1ca6808f2f1a2d80aed6cfce53fee 100644
--- a/data/10anhang/36whyoneoffsfail
+++ b/data/10anhang/36whyoneoffsfail
@@ -1,11 +1,11 @@
-<subsection title="How a oneOff network can become noninvertible" label="oofail">
+<subsection title="How a oneoff network can become noninvertible" label="oofail">
 
 
-The easiest model for understanding the oneOff width is something like #sqrt(abs(x-mean(x))**2+std(x)**2)#. And while the means usually match the training data, the standart deviation can be of any size. So training on a dataset and comparing it to another dataset with the same mean and less width, results in a noninvertible network. This is nothing we can do anything about, and an effect that is the same when we talk about autoencoder classifier, and is even less probable here, as we try to minimize the width, making it less probable that there is a distribution with lower width. Still this can happen (you can see the frequency in chapter <ref cross>), but here we want to mention one effect, that is even worse: antiinvertibility: if trained on a, b has lower loss, and if trained on b, a has lower loss. This is an ultra rare effect, as we have only observed it once (or multiple times if you think of the statistical invertibility of chapter <ref ldm>), and an effect that cannot happen just with an autoencoder, so how does this happen? In general, an oneoff network should not be able to do this, as if one feature has a certain width in a, and a lower feature in b, you should be able to pick the same feature in b resulting in b finding a more complicated, except for the case in which there is another feature in b with lower width, but this would also mean, that the width of the second feature in a would be bigger than of the first feature, since else it would have been choosen, resulting again in b finding a more complicated. Or in math: given #LessThan(f_1**b,f_1**a)# the network is not antiinvertible unless #LessThan(f_2**b,f_1**b)#, but since #LessThan(f_1**a,f_2**a)# it has also to be true that #LessThan(f_2**b,f_2**a)# so no network can be antiinvertible<note we simplify here a tiny bit, since you could mix two features, but this does not chance the math>. That beeing said, since we use autoencoder in the front, it can happen, that a feature of the first autoencoder just does not exist in the second one, thus breaking the logical chain, and making antiinvertible networks possible. We only ever saw a single event doing this, and it was a network working on ldm data (see chapter <ref ldm>). Ldm data is hard to differentiate at best, making noninvertible networks much more likely (in fact, as seen in chapter <ref cross>, all noninvertible oneOffs in this thesis are trained on ldm data), and by trying to scale using dense networks, at a node number of 25 we got a antiinvertible network
+The easiest model for understanding the oneoff width is something like #sqrt(abs(x-mean(x))**2+std(x)**2)#. And while the means usually match the training data, the standart deviation can be of any size. So training on a dataset and comparing it to another dataset with the same mean and less width, results in a noninvertible network. This is nothing we can do anything about, and an effect that is the same when we talk about autoencoder classifier, and is even less probable here, as we try to minimize the width, making it less probable that there is a distribution with lower width. Still this can happen (you can see the frequency in chapter <ref cross>), but here we want to mention one effect, that is even worse: antiinvertibility: if trained on a, b has lower loss, and if trained on b, a has lower loss. This is an ultra rare effect, as we have only observed it once (or multiple times if you think of the statistical invertibility of chapter <ref ldm>), and an effect that cannot happen just with an autoencoder, so how does this happen? In general, an oneoff network should not be able to do this, as if one feature has a certain width in a, and a lower feature in b, you should be able to pick the same feature in b resulting in b finding a more complicated, except for the case in which there is another feature in b with lower width, but this would also mean, that the width of the second feature in a would be bigger than of the first feature, since else it would have been choosen, resulting again in b finding a more complicated. Or in math: given #LessThan(f_1**b,f_1**a)# the network is not antiinvertible unless #LessThan(f_2**b,f_1**b)#, but since #LessThan(f_1**a,f_2**a)# it has also to be true that #LessThan(f_2**b,f_2**a)# so no network can be antiinvertible<note we simplify here a tiny bit, since you could mix two features, but this does not change the math>. That beeing said, since we use autoencoder in the front, it can happen, that a feature of the first autoencoder just does not exist in the second one, thus breaking the logical chain, and making antiinvertible networks possible. We only ever saw a single event doing this, and it was a network working on ldm data (see chapter <ref ldm>). Ldm data is hard to differentiate at best, making noninvertible networks much more likely (in fact, as seen in chapter <ref cross>, all noninvertible oneoffs in this thesis are trained on ldm data), and by trying to scale using dense networks, at a node number of 25 we got a antiinvertible network
 
-<i f="drantiinv25" f2="dsantiinv25" wmode="True">(doubleroc/sep)Antiinvertibility for ldm data, left using the autoencoder loss, and on rigth using oneOff networks</i>
+<i f="drantiinv25" f2="dsantiinv25" wmode="True">(doubleroc/sep)Antiinvertibility for ldm data, left using the autoencoder loss, and on rigth using oneoff networks</i>
 
 Interrestingly is the seperation quality here much better (even if reversed), as we will be able to do in chapter <ref ldm> with node sizes of 4. This we interpret that this difference is basically just the jet size, as the number of nodes show a difference between both ldm datapoints
 <i f="nhistldm" wmode="True">ldm size hist</i>
-and a network with less nodes (we tried 9 and 16, sizes that dont show many zero particles <note zero padded particles to be precise, for more information see chapter <ref data>>) does not show any real difference between both datasets. 36 nodes show a difference, but while not beeing invertible, this dataset is at least not antiinvertible, showing how rare those networks are.
+and a network with less nodes (we tried 9 and 16, sizes that don`t show many zero particles <note zero padded particles to be precise, for more information see chapter <ref data>>) does not show any real difference between both datasets. 36 nodes show a difference, but while not beeing invertible, this dataset is at least not antiinvertible, showing how rare those networks are.
 
diff --git a/data/10anhang/37oopar b/data/10anhang/37oopar
index 27000e52f1777ecc96ac60612de878d995c30152..a235cb0461bbe32a364fab6f56e21d75a09f5d3c 100644
--- a/data/10anhang/37oopar
+++ b/data/10anhang/37oopar
@@ -1,5 +1,5 @@
-<subsection title="Treating oneOff networks as observables" label="observable">
+<subsection title="Treating oneoff networks as observables" label="observable">
 
-If you use a very simple oneOff networks, on momentum 4 vectors (see appendix <ref oometrik>) results in this network learning the mass of the particles.
+If you use a very simple oneoff networks, on momentum 4 vectors (see appendix <ref oometrik>) results in this network learning the mass of the particles.
 This is interresting, since this is the same we would do to compare different particles by their 4 momenta and we now have an algorithm to automatically extract this feature just from data. So it migth be fair to assume that by applying this algorithm to more complicated data, we still extract some feature, and that we can use this feature to find anomalies. But giving a feature, you can do more: you can look at statistical information: if you only produce electrons in you detector, but you measure masses that are on average a bit higher than #500*keV#, you can conclude that their is something else produced but just electrons. The benefit here is, that you can combine multiple events to get lower uncertainities on the variable you care about, and thus you can easier detect irregularities in your dataset. So when we have an automatic feature extractor, it migth be interresting to see if you can differentiate between datasets using this feature.
-We use here #1000000# jets, of which #0.01# are not QCD but top jets<note this is about the most we can do giving our dataset>. This is enough to reach a significance of #4.6# sigma on a single oneOff network<note no combination of multiple runs>. So oneOff features migth be appliable to finding new physics. Most interrestingly this probably be applied to any dataset, and so you could define detectorlevel features to directly compare your data to the expectation, assuming your simulation are good enough.
\ No newline at end of file
+We use here #1000000# jets, of which #0.01# are not QCD but top jets<note this is about the most we can do giving our dataset>. This is enough to reach a significance of #4.6# sigma on a single oneoff network<note no combination of multiple runs>. So oneoff features migth be appliable to finding new physics. Most interrestingly this probably be applied to any dataset, and so you could define detectorlevel features to directly compare your data to the expectation, assuming your simulation are good enough.
\ No newline at end of file
diff --git a/data/10anhang/38lpt b/data/10anhang/38lpt
index 1758412f97a1a2dabd19277fab1fea082c0b04f3..66527a7b737efa783a47a736478eb7e3b900266f 100644
--- a/data/10anhang/38lpt
+++ b/data/10anhang/38lpt
@@ -1 +1 @@
-<subsection title="Chancing the definition of the transverse momentum" label="alpt">
+<subsection title="changing the definition of the transverse momentum" label="alpt">
diff --git a/data/10anhang/42comparepnet b/data/10anhang/42comparepnet
index 815eb2d15774a49c2846bedb386f42c5aafe72d3..1f1e3e8b0d9fba8015b4cf23413b431495bd1ed1 100644
--- a/data/10anhang/42comparepnet
+++ b/data/10anhang/42comparepnet
@@ -1,5 +1,5 @@
 <subsection title="Comparing our graph update layer to particleNet" label="acomparepnet">
 
 
-There are multiple different ways of implementing such a layer, a notable one is the one used by particleNet <cite particleNet>: Their graph connectivity is implemented, by just storing all neighbouring vector to each given vector in a set of vectors, this means, they can implement the update procedure as a function of the original and the neighbour vectors<note this function is actually a bit complicated, involving not only convolutions, but also normalisations between them, and they end by concatting the updated vector to the original one, which is something that is not very useful, when you want to reduce the size of your graph>. This is not exactly what we do here, mostly since the implementation of the graph as just a corresponding set of neighbourvectors demands for computational reasons that each node is connected to a same number of other nodes, and also requires relearning your graph after each step, which we dont want to force our network to do, as explained in chapter <ref arelearn>, and also would make this less of a graph autoencoder, and more into an autoencoder with some graph update layers in front of it (which migth also not be a good idea, see chapter <ref ainfront>), since there is no way to reduce the number of nodes for such an implementation, without completely ignoring the graph structure.
+There are multiple different ways of implementing such a layer, a notable one is the one used by particleNet <cite particleNet>: Their graph connectivity is implemented, by just storing all neighbouring vector to each given vector in a set of vectors, this means, they can implement the update procedure as a function of the original and the neighbour vectors<note this function is actually a bit complicated, involving not only convolutions, but also normalisations between them, and they end by concatting the updated vector to the original one, which is something that is not very useful, when you want to reduce the size of your graph>. This is not exactly what we do here, mostly since the implementation of the graph as just a corresponding set of neighbourvectors demands for computational reasons that each node is connected to a same number of other nodes, and also requires relearning your graph after each step, which we don`t want to force our network to do, as explained in chapter <ref arelearn>, and also would make this less of a graph autoencoder, and more into an autoencoder with some graph update layers in front of it (which migth also not be a good idea, see chapter <ref ainfront>), since there is no way to reduce the number of nodes for such an implementation, without completely ignoring the graph structure.
 Please note the difference: Since we use an adjacency matrix itself to define the graph(in comparison to calculating some derivative from it), you not only have complete control over the graph, that can be used to shrink the graph structure with the number of feature vectors, but you also allow for an arbitrary number of connections for each node<note this is mostly interresting, since it extends the number of possible compression algorithms: They do not anymore have to satisfy keeping the number of connections constant: The number of possible graphs with #n# nodes is #2**(n*(n-1)/2)# (ignoring permutation invariance, self connectivity and directed graphs), for #Eq(n,4)# this results in #64# possible graphs, of which only #6# are of this kind. This means that much less compressed graphs are possible, and that finding an algorithm, that can pick only those graphs, is much more complicated (see appendix <ref atopkwhy> for more)><ignore><note you migth also have noticed, that there is another difference, since my implementation does not allow for cross terms between the self and neighbour terms. This is just a minor difference, since in the following graph update step, this is still given, and you would expect them to take a less importatn role anyway></ignore>.
\ No newline at end of file
diff --git a/data/10anhang/60sorting b/data/10anhang/60sorting
index 49dfb66423d64e5ae59380cbe23298d648e1d0ed..0b287a0f13c300957f0b7d0b8f9b35097a094c64 100644
--- a/data/10anhang/60sorting
+++ b/data/10anhang/60sorting
@@ -4,6 +4,6 @@ Sorting nodes at the end of the autoencoder breaks the permutation symmetry that
 <i f="vis928" f2="vis1025" wmode="True">sorting in network plot comparison</i>
 The sorted network reduces an AUC value (trained on QCD) of #0.6351# into #0.5788# and probably even more importantly, the training curve looks way worse
 <i f="history928" f2="history1025" wmode="True">training curve comparison for sorting</i>
-You migth consider the not sortet curve more clean, but it also does not really improve any further at a fairly early epoch, result in the sorting network reaching a loss of about a factor 3 smaller. Seeing this, sorting seems like a clear choice for us, theoretical concerns dont fair well compared to practical results, and so basically all networks in this thesis are sorted. That beeing said, the original deficits of breaking permutations symmetry, could actually be interpreted to mean the opposite: while it is true, that switching each value, except the sorted one, would not result in the same loss, switching any whole node position, still would result in exactly the same loss. In fact, we can use this, to understand why not sorted networks are so bad: The encoding includes a random<note actually not random, but you could see it like this, if you only see the initial and final node indices> node permutation, while the decoding does not, so after the autoencoder, the result is a random permuation of the input features, which then are compared to the still initially ordered input features. That this does not work that well should be clear: So either choose the momentum axis as sorting value<note which would not be what you would want, since locality in real space is much more important than similar energies, as chapter <ref ametrikana> shows> or compare your predictions neirly randomly. In fact, you could argue, that this breaks permutation symmetry, as you impose a defined ordering on your node indices. Finally, if you would want to improve this, you migth look at two things: making the comparison variable learnable, would remove the artificial inflated importance of #lp_T#<note but maybe also make the training less stable, and add a less controlable importance to some other mixture of features> and making the decompression chance the node ordering, best case in a learnable way, would make this whole discussion moot, as the network could converge as good with, as without sorting. That beeing suggested, implementing this is not neccesarily easy, as you would not want any function to apply to all nodes, to make sure you dont break permutation symmetry, which for me looks like you restrict yourself to finding a variable to sort by and to reverse an initial sorting would in general not be easy at all, as the initial sorting could be completely random, but would result here probably in a network sorting the nodes by their transverse momentum, as this is the sorting of the initial data, but this seems to us, as a more complicated implementation of our final sorting layer<note that could actually work less well, not only since it needs more calculcation time, but also since this sorting is done at deconstruction, meaning that later graph update layer wont have an effect on it><note you migth also ask yourself if you could not just remove the permutation from the encoding layer, but this is easier said than done: As it is true, that the sorting is generally just done for implementation, but as you combine 4 values into for example two, you could have situations, in which node 0 and 3, as well as node 1 and 2 are combined together into (0,3) and (1,2), and even without sorting, reconstruction this, would result in 0,3,1,2, so you would still either need some kind of permutation in the decompression, or some kind of shortcuts between the layers, that encodes the original position: This would not be bad style, as it could result in the network learning to misuse this information to encode arbitrary information, but would also not be very easy to implement, and migth require a nonpermutation invariant compression and decompression function to work well, which would obviously not be ideal, as keeping permutation invariance is the main reason for this chapter>.
-So finally: sorting seems to be the rigth choice for us, but a more advanced algorithm, migth still be useful: consider the data from chapter <ref feyn>: sorting by one parameter is not that useful, when you only have boolean datapoints<note even though in this chapter no real sorting was used, and you could still work with our approach and multiple sorting layers fairly well, assuming tf.math.top\_k is stable (their documentation does not say so, but the implementation is, but this may chance since also tf.argsort is stable at the time of writing this, but they want to implement a not stable version later to improve the speed of this algorithm)>
+You migth consider the not sortet curve more clean, but it also does not really improve any further at a fairly early epoch, result in the sorting network reaching a loss of about a factor 3 smaller. Seeing this, sorting seems like a clear choice for us, theoretical concerns don`t fair well compared to practical results, and so basically all networks in this thesis are sorted. That beeing said, the original deficits of breaking permutations symmetry, could actually be interpreted to mean the opposite: while it is true, that switching each value, except the sorted one, would not result in the same loss, switching any whole node position, still would result in exactly the same loss. In fact, we can use this, to understand why not sorted networks are so bad: The encoding includes a random<note actually not random, but you could see it like this, if you only see the initial and final node indices> node permutation, while the decoding does not, so after the autoencoder, the result is a random permuation of the input features, which then are compared to the still initially ordered input features. That this does not work that well should be clear: So either choose the momentum axis as sorting value<note which would not be what you would want, since locality in real space is much more important than similar energies, as chapter <ref ametrikana> shows> or compare your predictions neirly randomly. In fact, you could argue, that this breaks permutation symmetry, as you impose a defined ordering on your node indices. Finally, if you would want to improve this, you migth look at two things: making the comparison variable learnable, would remove the artificial inflated importance of #lp_T#<note but maybe also make the training less stable, and add a less controlable importance to some other mixture of features> and making the decompression change the node ordering, best case in a learnable way, would make this whole discussion moot, as the network could converge as good with, as without sorting. That beeing suggested, implementing this is not neccesarily easy, as you would not want any function to apply to all nodes, to make sure you don`t break permutation symmetry, which for me looks like you restrict yourself to finding a variable to sort by and to reverse an initial sorting would in general not be easy at all, as the initial sorting could be completely random, but would result here probably in a network sorting the nodes by their transverse momentum, as this is the sorting of the initial data, but this seems to us, as a more complicated implementation of our final sorting layer<note that could actually work less well, not only since it needs more calculcation time, but also since this sorting is done at deconstruction, meaning that later graph update layer wont have an effect on it><note you migth also ask yourself if you could not just remove the permutation from the encoding layer, but this is easier said than done: As it is true, that the sorting is generally just done for implementation, but as you combine 4 values into for example two, you could have situations, in which node 0 and 3, as well as node 1 and 2 are combined together into (0,3) and (1,2), and even without sorting, reconstruction this, would result in 0,3,1,2, so you would still either need some kind of permutation in the decompression, or some kind of shortcuts between the layers, that encodes the original position: This would not be bad style, as it could result in the network learning to misuse this information to encode arbitrary information, but would also not be very easy to implement, and migth require a nonpermutation invariant compression and decompression function to work well, which would obviously not be ideal, as keeping permutation invariance is the main reason for this chapter>.
+So finally: sorting seems to be the rigth choice for us, but a more advanced algorithm, migth still be useful: consider the data from chapter <ref feyn>: sorting by one parameter is not that useful, when you only have boolean datapoints<note even though in this chapter no real sorting was used, and you could still work with our approach and multiple sorting layers fairly well, assuming tf.math.top\_k is stable (their documentation does not say so, but the implementation is, but this may change since also tf.argsort is stable at the time of writing this, but they want to implement a not stable version later to improve the speed of this algorithm)>
 
diff --git a/data/10anhang/75uncertain b/data/10anhang/75uncertain
index e87317a8d0097832947d4673ba661f5c1959df97..945b109b7bae88f8b2023399d5857d4ced3c441e 100644
--- a/data/10anhang/75uncertain
+++ b/data/10anhang/75uncertain
@@ -1,7 +1,7 @@
 <subsection title="Uncertainity and reproducability of AUC values" label="auncertain">
 
 
-I am not a big fan of measuring uncertainities for neural networks, by just repeating the training phase and comparing the output, because as chapter (ENTER LINK) shows, we can reduce this uncertainity neirly arbitrarily long, by just training more carefully. Even though this is the reason why most values in this thesis dont have errors attached to them, here is a simple reproducability study
+I am not a big fan of measuring uncertainities for neural networks, by just repeating the training phase and comparing the output, because as chapter (ENTER LINK) shows, we can reduce this uncertainity neirly arbitrarily long, by just training more carefully. Even though this is the reason why most values in this thesis don`t have errors attached to them, here is a simple reproducability study
 
 <i f="none" wmode="True">reproducability network setup</i>
 
diff --git a/data/10anhang/80graphrelearning b/data/10anhang/80graphrelearning
index 2961a9d17d47a5b8b243ea5b5abc672c55583f80..bf8ea5099913876f690f19e5b01e453a6a4cccac 100644
--- a/data/10anhang/80graphrelearning
+++ b/data/10anhang/80graphrelearning
@@ -3,7 +3,7 @@
 
 Quick answer: No.
 Long answer: Probably no, but not because the quality is neccesarily worse, just because the number of nans (chapter <ref nan>) increases a lot, making training for a long time very hard and thus resulting in worse classifiers.
-That beeing said, this still means, that if you could handle the nans, you migth profit from more gtopk layer, but we are not able to test this at the moment, and even though multiple different graphs help interpreting graphs as activations (chapter <ref agaeactivation>), there is not really any physically useful definition of similarity in angles and momenta, but the angles themself, so chancing the graph setup in the middle of the layers, migth not have any effect at all.
+That beeing said, this still means, that if you could handle the nans, you migth profit from more gtopk layer, but we are not able to test this at the moment, and even though multiple different graphs help interpreting graphs as activations (chapter <ref agaeactivation>), there is not really any physically useful definition of similarity in angles and momenta, but the angles themself, so changing the graph setup in the middle of the layers, migth not have any effect at all.
 
 
 
diff --git a/data/10anhang/85sizematters b/data/10anhang/85sizematters
index 08ddcb017e2d069c3b69fb01db5c8d8fa14e38cd..9c8826b2bdd3330569fcb9221c3363e6ec85818a 100644
--- a/data/10anhang/85sizematters
+++ b/data/10anhang/85sizematters
@@ -1,7 +1,7 @@
-<subsection title="Trainingsize, and why graph autoencoder dont care about it" label="asize">
+<subsection title="Trainingsize, and why graph autoencoder don`t care about it" label="asize">
 
 
-Graph networks show some properties that usual network dont. One thing is the apparent independence of the training size, and even when you can easily explain this, as having few parameters in your network, this still migth allow you to train on data that was not usable before (see chapter <ref secuse>)
+Graph networks show some properties that usual network don`t. One thing is the apparent independence of the training size, and even when you can easily explain this, as having few parameters in your network, this still migth allow you to train on data that was not usable before (see chapter <ref secuse>)
 
 <i f="none" f2="none" f3="none" wmode="True"> loss curves for 3 training sizes</i>
 <i f="none" wmode="True">training size triple roc curve</i>
diff --git a/data/11otheruses/02nets b/data/11otheruses/02nets
index 0ab07fcebd41b79fa227dc61ea45ed44bbb1f2c4..9f80d590210799aaf6722ecc8538e18eb4561cd6 100644
--- a/data/11otheruses/02nets
+++ b/data/11otheruses/02nets
@@ -5,7 +5,7 @@ The corresponding tutorial can be found here (ENTER LINK) and the full code is f
 
 <subsubsection title="Datageneration" label="netsdata">
 Datageneration is often the most timeconsuming part of a new neural network, and it would not be different here. So to save some time, we just generate a sample social network. This allows you to ignore privacy settings<you migth not know everything about every user: you would need to decide how to handle a friend about whom you do not know some critical information>, simplifies the problem a bit<note since an usual facebook user has a lot of information, and often enough hundrets of friends>, and allows you to clearly define the anomalous data points. That beeing said, this also means, that we could tweak the data in every possible way to make the results arbitrarily good, which is also why this is the only subchapter that works with self generated data.
-This generated network consists of 5000 randomly generated users with 4 attributes (a constant 1 (flag), #a#:an integer between 1 and 3, #b#:an integer either 0 or 1 as well as a normal distributed value that depends on #a#<note a normal distributed value with mean #0# and standart deviation #1# added to #(2**a)/16# times another normal distribution with mean #1# and standart deviation #0.1#. This is done just to have some relation between the elements>. The corresponding connections are generated the following way: each connection has a probability, that depends on the difference in the person vector<note a factor #exp(-abs(x_i-x_j)**2)#> and on the difference in the node index<note another factor #exp(-0.1*abs(i-j))#>. This means, that more similar persons are connected more closely, and that friends of friends are more probably friends. Now we guess on average 5 connections for each person, with respect to the given probabilities, or 2 for the alternative datapoints. We choose these anomalies, since defining less used accounts as signals allows us later to show a benefit of oneOff networks.
+This generated network consists of 5000 randomly generated users with 4 attributes (a constant 1 (flag), #a#:an integer between 1 and 3, #b#:an integer either 0 or 1 as well as a normal distributed value that depends on #a#<note a normal distributed value with mean #0# and standart deviation #1# added to #(2**a)/16# times another normal distribution with mean #1# and standart deviation #0.1#. This is done just to have some relation between the elements>. The corresponding connections are generated the following way: each connection has a probability, that depends on the difference in the person vector<note a factor #exp(-abs(x_i-x_j)**2)#> and on the difference in the node index<note another factor #exp(-0.1*abs(i-j))#>. This means, that more similar persons are connected more closely, and that friends of friends are more probably friends. Now we guess on average 5 connections for each person, with respect to the given probabilities, or 2 for the alternative datapoints. We choose these anomalies, since defining less used accounts as signals allows us later to show a benefit of oneoff networks.
 Now for each person in this network, we only look at the local surrounding of this person. This is done, by taking only the connection of the friends, or friends of friends of this person into account. This generates a lot smaller graphs, that we can now feed into the autoencoder<note you could ask yourself if this reusing of nodes does not result in a lot of overfitting (by learning the nodes themself), but as you see below, that is not the case, possibly because of the low number of parameters in the graph autoencoder>, but for simplicity we cut a bit on the size of those new graphs, as we allow for at most 70 nodes<note this still keeps #0.9932# of all data points>.
 
 <subsubsection title="Training" label="netstrain">
@@ -14,9 +14,9 @@ To train this network, we use a fairly simple setup, compressing the 70 nodes on
 As you see in the training curve, the loss is basically the same for trainings and validation loss, and contains steps, as also seen before<note for example in chapter <ref secondworking>>.
 Much more interestingly, is the loss distribution
 <i f="histlossNETS" wmode="True">loss distribution for this network</i>
-As you see, the reconstruction is not very good, as basically all events have a nonzero loss, but maybe even more interrestingly: there is some difference in the reconstruction of out anomal datapoints. This migth seems like you could use this to seperate datapoints, but there is a difficulty: If you use an autoencoder to seperate datasets, you assume that a dataset which the network never saw, will be reconstructed worse than a dataset that is trained on, but here the opposite is the case: the data that is abnormal is easier reconstructed<note this reminds of of the case of nonnormated nets trained on top jets>, so any seperation is a bit weird: you could just look at something like #1-loss#, but since you do not have any reasoning for this this makes the training no longer unsupervised. Also probably only this kind of abnormal data will be reconstructed easier<note this is here probably the case, sine the abnormal data is less complicated, as it contains less nodes, you migth be able to handle this, by defining your loss relative to the number of nodes, but this misses the point a bit, as more easy anomalies can still exist, and we can show that oneOff networks can handle them>, and by negating the loss, you would not get any useful seperation on other datapoints. So what can we do? Use oneOff networks: In their easiest version, they take the mean of the training peak, and define distance as difference to this peak, which would already solve this problem, and in their deep implementation they migth even improve this further. Anyhow why, this works quite well
+As you see, the reconstruction is not very good, as basically all events have a nonzero loss, but maybe even more interrestingly: there is some difference in the reconstruction of out anomal datapoints. This migth seems like you could use this to seperate datapoints, but there is a difficulty: If you use an autoencoder to seperate datasets, you assume that a dataset which the network never saw, will be reconstructed worse than a dataset that is trained on, but here the opposite is the case: the data that is abnormal is easier reconstructed<note this reminds of of the case of nonnormalized nets trained on top jets>, so any seperation is a bit weird: you could just look at something like #1-loss#, but since you do not have any reasoning for this this makes the training no longer unsupervised. Also probably only this kind of abnormal data will be reconstructed easier<note this is here probably the case, sine the abnormal data is less complicated, as it contains less nodes, you migth be able to handle this, by defining your loss relative to the number of nodes, but this misses the point a bit, as more easy anomalies can still exist, and we can show that oneoff networks can handle them>, and by negating the loss, you would not get any useful seperation on other datapoints. So what can we do? Use oneoff networks: In their easiest version, they take the mean of the training peak, and define distance as difference to this peak, which would already solve this problem, and in their deep implementation they migth even improve this further. Anyhow why, this works quite well
 <i f="oohistNETS" wmode="True">oo dist for nets</i>
-Please note that we dispence of using any number here, measuring how good the reconstruction is, as we could improve it arbitrarily by chancing the data generation
+Please note that we dispence of using any number here, measuring how good the reconstruction is, as we could improve it arbitrarily by changing the data generation
 
 <subsubsection title="Whats next" label="netsnext">
 Given these examples, you migth notice, that they are not though through completely. This is why we include these kind of subchapter to give you some ideas on what could be improved further. The first thing you migth need, is to work with more kinds of anomalous datapoints, and not just neglected profiles. It migth also be a good idea to work on an actual social network, and if you do this, it would be interresting to just look at the users in the training set, that are reconstructed worst, as this would allow you to find abnormal users in a truly unsupervised way.
diff --git a/data/11otheruses/03mol b/data/11otheruses/03mol
index b795ce0c6d50b7dc063f1d57ec68b55ad8b7240d..3ee57c667bbca4ac1a8d94955aa9a650bbde822f 100644
--- a/data/11otheruses/03mol
+++ b/data/11otheruses/03mol
@@ -13,10 +13,10 @@ Those datapoints, get filtered a lot, since fewer events do not really matter as
 <i f="visMOL1" f2="visMol2" wmode="True">modelsetup, with or without graph compression step</i>
 We again use a fairly simple setup, consisting only out of a handful of graph update layer, and possibly a graph compression layer, comparing its effect
 <i f="historyMOL" f2="historyMOL2" wmode="True">training results mol 5/6</i>
-There are two things to note here: first, since the mass can reach order of magnitude of #1000*g/mol#, and since the difference is squared, this can reach very high loss values at the beginning of the training. Combine this with the fact, that masses are very easy to predict, and this is why you see orders of magnitude of chance in the loss function. As you also see, the chance in loss is different between the compressing network, and the noncompressed version: As both networks require similar times to calculate an training epoch, the compressed version requires more than 100 epochs less to reach a similar result. That beeing said, therefore the noncompressed version reaches a sligthly lower minimal loss of #78# in comparison to #73#, even though it should be noted, that this difference is tiny compared to initial losses, and the compressing version has a bunch more parameters that migth be able to be tweaked to chance this.
+There are two things to note here: first, since the mass can reach order of magnitude of #1000*g/mol#, and since the difference is squared, this can reach very high loss values at the beginning of the training. Combine this with the fact, that masses are very easy to predict, and this is why you see orders of magnitude of change in the loss function. As you also see, the change in loss is different between the compressing network, and the noncompressed version: As both networks require similar times to calculate an training epoch, the compressed version requires more than 100 epochs less to reach a similar result. That beeing said, therefore the noncompressed version reaches a sligthly lower minimal loss of #78# in comparison to #73#, even though it should be noted, that this difference is tiny compared to initial losses, and the compressing version has a bunch more parameters that migth be able to be tweaked to change this.
 
 <subsubsection title="Whats next?" label="molnext">
-We dont want to call using a compression layer to pool graph networks generally a good idea, but if you have a network that takes an unbearable time, trying out inserting a compression layer migth be good idea. It migth also be interresting to optimize the hyperparameters of the compression layer, or even to alter the setup by for example using an abstraction layer. Finally, this is tested on a fairly easy setup, and it migth be interresting to use this on a more complicated setup like particleNet.
+We don`t want to call using a compression layer to pool graph networks generally a good idea, but if you have a network that takes an unbearable time, trying out inserting a compression layer migth be good idea. It migth also be interresting to optimize the hyperparameters of the compression layer, or even to alter the setup by for example using an abstraction layer. Finally, this is tested on a fairly easy setup, and it migth be interresting to use this on a more complicated setup like particleNet.
 
 
 
diff --git a/data/11otheruses/04feyn b/data/11otheruses/04feyn
index 57422ac440199c4258c957eab10f97179744c950..7647a0cf054e465bd1d8a6dc5a5bc96aae71f130 100644
--- a/data/11otheruses/04feyn
+++ b/data/11otheruses/04feyn
@@ -1,18 +1,18 @@
 <subsection title="High level machine learning and feynman diagramms<ignore>, or how i learned to stop thinking and love the graph</ignore>" label="feyn">
 
-Machine learning and anomaly detection is usually only used on low level data. Inputs that are easily generated but timeconsuming for humans to understand. But why not apply machine learning to highly abstracted concepts? You migth ask why one would want this: One result migth be something like a theory evaluation method: If you have a number of predictions, this could classify weirdness in the sense of finding predictions that dont match the rest. In the best case you could also extend theories consistently: you can generate new inputs from existing ones. You could automatically bring structure to your predictions, by looking at the compression space of an autoencoder or you could use this to simplify complicated theories. So why dont we do this? two things come to mind: most theories can not be brougth into vector form, and generating a lot of predictions is quite hard. Luckely both are solved by the graph setup: This graph structure is way more powerful, to the point that artificial intelligence research often encodes knowledge in graphs <ignore>(see (ENTER REFERENCE))</ignore>, and since overfitting has not been a problem at all here, also the low number of training samples should not matter here<note there is a second price you pay, when you train on a few datapoints: Not only becomes overfitting more probable, but you also loose generality, as density fluctuations of the different kind of training samples (where these types of samples are defined by the training itself, which makes them hard to filter out) start to matter more. Sadly we cannot really chance this to much>
+Machine learning and anomaly detection is usually only used on low level data. Inputs that are easily generated but timeconsuming for humans to understand. But why not apply machine learning to highly abstracted concepts? You migth ask why one would want this: One result migth be something like a theory evaluation method: If you have a number of predictions, this could classify weirdness in the sense of finding predictions that don`t match the rest. In the best case you could also extend theories consistently: you can generate new inputs from existing ones. You could automatically bring structure to your predictions, by looking at the compression space of an autoencoder or you could use this to simplify complicated theories. So why don`t we do this? two things come to mind: most theories can not be brougth into vector form, and generating a lot of predictions is quite hard. Luckely both are solved by the graph setup: This graph structure is way more powerful, to the point that artificial intelligence research often encodes knowledge in graphs <ignore>(see (ENTER REFERENCE))</ignore>, and since overfitting has not been a problem at all here, also the low number of training samples should not matter here<note there is a second price you pay, when you train on a few datapoints: Not only becomes overfitting more probable, but you also loose generality, as density fluctuations of the different kind of training samples (where these types of samples are defined by the training itself, which makes them hard to filter out) start to matter more. Sadly we cannot really change this to much>
 Now consider feynman diagramms: As they are able to encode particle physics in a finite set of graphs, they are at the same time very high level, while also still providing #O(100)# samples, which should be barely enough for us to train on, and finding anomalous feynman diagrams migth actually be an interresting way to solve this thesis initial idea of using graphs to find new physics
 
 The corresponding tutorial can be found here (ENTER LINK) and the full code is found here (ENTER LINK)
 
 <subsubsection title="Data generation" label="feyndata">
-Datageneration for feynman diagrams means more converting data, instead of outrigth generating them. The problem is, that all diagrams that you find, are usually given as images, and writing an program to read every image into a diagram is absolutely nontrivial, which is why we just converted those diagramms by hand<note you could actually use a graph neural network for this, build similar to the one from the next chapter <ref build>>. That beeing said, you could actually ask yourself, if writing an image like autoencoder to work on those images would not be much less work. And even though we would agree, we think this would also work way worse, as you could not differentiate between an image that just looks like a feynman diagram, and an image that actually represents some physical insigth<note you see this quite clearly in another usecase we were thinking about: Recipes are easily generated by a text based gan, or better texts that look like recipes, but generating recipes that actually taste good is much harder, and you dont really have a way to test this(beside cooking for a long time), as your loss could also just say how much your text looks like a recipe>. If everything looks like a feynman diagramm, you can easily use the loss to differentiate those two cases, since a chance in loss now definitely represents a better reconstruction in the autoencoder we will train. Also by training on images you could again more probably see overfitting, resulting in higher needed training samples, that we dont have.
+Datageneration for feynman diagrams means more converting data, instead of outrigth generating them. The problem is, that all diagrams that you find, are usually given as images, and writing an program to read every image into a diagram is absolutely nontrivial, which is why we just converted those diagramms by hand<note you could actually use a graph neural network for this, build similar to the one from the next chapter <ref build>>. That beeing said, you could actually ask yourself, if writing an image like autoencoder to work on those images would not be much less work. And even though we would agree, we think this would also work way worse, as you could not differentiate between an image that just looks like a feynman diagram, and an image that actually represents some physical insigth<note you see this quite clearly in another usecase we were thinking about: Recipes are easily generated by a text based gan, or better texts that look like recipes, but generating recipes that actually taste good is much harder, and you don`t really have a way to test this(beside cooking for a long time), as your loss could also just say how much your text looks like a recipe>. If everything looks like a feynman diagramm, you can easily use the loss to differentiate those two cases, since a change in loss now definitely represents a better reconstruction in the autoencoder we will train. Also by training on images you could again more probably see overfitting, resulting in higher needed training samples, that we don`t have.
 We use all diagrams from <cite diagramms><note these diagramms are of relatively low order>, that match our filter of only SM diagrams and at most 9 lines, and represent each diagram in the following way: Each line becomes a node, and each two lines that meet in an edge, are connected. This migth seem counterintuitive at first, as we basically switch nodes and edges, but is actually neccesary, since each edge requires two nodes, and in most usual feynman diagramm this is not given, as input aswell as output lines, only have one edge. Then each line(node) is represented by a 14 dimensional vector, onehot<note onehot encoding means encoding a number that is smaller than #a# by a vector of values which element #i# is #1# is the number is #i# and #0# else> encoding the particle type (gluon,quark,lepton,muon,Higgs, W Boson, Z Boson, photon, proton,jet), 3 special boolean values encoding anti particles<note for simplicity this variable is always zero for lines that are neither input nor output>, input lines, output lines and a fourtheenth value that is 1 (similar to flag (see chapter <ref data>)).
 <i f="conv01" f2="conv02" wmode="True"> Example image of the conversion, transforming left into rigth </i>
 
 <subsubsection title="Training" label="feyntrain">
 <i f="visfeyn" wmode="True">Model setup plot</i>
-Also here a fairly easy setup is used, but instead of the compression algorithm, we use the abstraction one, and the paramlike deconstruction algorithm replaces the classical one, to encode the abstraction of a factor 3 (reducing 9 nodes into 3). Therefore we add 3 parameters, as well as a couple more graph update steps. One thing that migth be important later, is that we dont punish the resulting graph structure directly, even though the paramlike decompression algorithm should make this possible, but only indirectly through the fact that a nonsencical graph structure will worsen the quality of the update step.
+Also here a fairly easy setup is used, but instead of the compression algorithm, we use the abstraction one, and the paramlike deconstruction algorithm replaces the classical one, to encode the abstraction of a factor 3 (reducing 9 nodes into 3). Therefore we add 3 parameters, as well as a couple more graph update steps. One thing that migth be important later, is that we don`t punish the resulting graph structure directly, even though the paramlike decompression algorithm should make this possible, but only indirectly through the fact that a nonsencical graph structure will worsen the quality of the update step.
 <i f="historyfeyn" wmode="True">Training history plot</i>
 As you see, the training curve improves after the initial plateau first quite drastically, just to slow down later, and reach a validation loss below #0.05# at the end, which we are fairly happy with, since this means, that converted to booleans, only about 1 in 20 values is wrong<note since the results are not booleans, this is only true for the average>. More interrestingly, you also see, that the validation loss is consistently lower than the training loss, which means, that even this network does not overfit and we thus migth be able to train a network on only #O(100)# events.
 <i f="lossbynFEYN" wmode="True">Loss of each training graph compared to its number of lines (IGNORE THE YELLOW POINTS HERE)</i>
@@ -33,7 +33,7 @@ These six diagramms migth suggest that this method works, but at the end, these
 this migth suggest, that weigthing the adjacency matrix directly would be a good idea. you migth also want to take a look at permutatation invariant losses (see chapter <ref losses>). Secondly, most diagramms have two inputs, and the network is fairly good at reconstructing them
 <i f="histicFEYN" wmode="True">input number hist</i>
 As you see, it migth even be a bit to good, as it reconstructs even more 2 input diagramms. (SEARCH FOR ERRORS IN THOSE RECONSTRUCTE DIAS)
-Finally reproducability and the applicability of oneOff networks migth also be interresting here.
+Finally reproducability and the applicability of oneoff networks migth also be interresting here.
 
 
 
diff --git a/data/11otheruses/05build b/data/11otheruses/05build
index 5002f0a2534484533c2b0a6a699c89c456c2d363..048d7719fe5b68c67bbb8951961e605fe617636f 100644
--- a/data/11otheruses/05build
+++ b/data/11otheruses/05build
@@ -10,7 +10,7 @@ As we had an examples for autoencoder, and an example for a supervised graph net
 The corresponding tutorial can be found here (ENTER LINK) and the full code is found here (ENTER LINK)
 
 <subsubsection title="Data generation" label="builddata">
-Data is here the biggest problem, as neither can we simply generate building plans ourself, nor is their any database of building plans, that we can use to generate graphs, or at least we did not find any. So basically we dont have any data, until somebody manually translates hundrets of building plans, into any computer readable format, and since we neither have the expertise nor the time to do this, the original idea of using a generative adversial network to generate new buildings is basically impossible. To be able to still give an example of graph generating networks, we replace one of the networks of the GAN by a predefined algorithm. This migth seem, like you can design arbitrary buildings, by just defining what you want, but it has some big problems, because of which we would still suggest the GAN approach more: first, defining what makes a building into a good building is not at all easy. In fact, in the following we just hope for orthogonal walls, by setting the loss to sum over all connections of #abs(x_1*x_2)# in the hope of generating walls that are parallel to the axis, and thus orthogonal to each other. Other than the fact that this loss is really basic, and we would not even expect any nice outputs, even this loss is so complicated, that tensorflow needs about as long to generate the differentials, as it needs to train the whole network, which is also why we choose to reduce the problem to two dimensions. Finally we add some loss to assert that the width of the building is fixed.
+Data is here the biggest problem, as neither can we simply generate building plans ourself, nor is their any database of building plans, that we can use to generate graphs, or at least we did not find any. So basically we don`t have any data, until somebody manually translates hundrets of building plans, into any computer readable format, and since we neither have the expertise nor the time to do this, the original idea of using a generative adversial network to generate new buildings is basically impossible. To be able to still give an example of graph generating networks, we replace one of the networks of the GAN by a predefined algorithm. This migth seem, like you can design arbitrary buildings, by just defining what you want, but it has some big problems, because of which we would still suggest the GAN approach more: first, defining what makes a building into a good building is not at all easy. In fact, in the following we just hope for orthogonal walls, by setting the loss to sum over all connections of #abs(x_1*x_2)# in the hope of generating walls that are parallel to the axis, and thus orthogonal to each other. Other than the fact that this loss is really basic, and we would not even expect any nice outputs, even this loss is so complicated, that tensorflow needs about as long to generate the differentials, as it needs to train the whole network, which is also why we choose to reduce the problem to two dimensions. Finally we add some loss to assert that the width of the building is fixed.
 
 <subsubsection title="Training" label="buildtrain">
 <i f="none" wmode="True">network setup</i>
diff --git a/data/old/01.txt b/data/old/01.txt
index 945c27ec98694a5766193d43080df236739c8856..fa2fbfa967237e4a7c591f9b67e15b87ed9453c3 100644
--- a/data/old/01.txt
+++ b/data/old/01.txt
@@ -1,13 +1,13 @@
 <ignore>
-I did not spend much (enough) time making thinks sound nice, and i dont think i will use handwritten graphs (but the libraries to plot graphs are kinda terrible and i think its ok for now)
+I did not spend much (enough) time making thinks sound nice, and i don`t think i will use handwritten graphs (but the libraries to plot graphs are kinda terrible and i think its ok for now)
 <section about decoders>
 
 <subsection mathematical description of a graph>
-To work numerically with a graph, you can use the adjacency matrix of it, its entry #A_ij# are #1# if there is a connection between the nodes #i# and #j#<note this is a bit of a simplification, since the order of nodes does not matter, You can say that multiplication with any permutationmatrix does not neccesarily keep the adjacency matrix invariant, but still represents the same graph. I will try order my graphs from left to rigth or i`ll enumerate them> and #0# if not. You can extend this by replacing the #1# with a value defining the strength of a connection or by transforming the graph into a directed graph by removing the symmetry of #A# (#A_ij# is one exactly if there is a connection from #i# to #j#), but both of those extensions do not chance much when we talk about decompression. Finally, there are the diagonal entries #A_ii# of the adjacency matrix, that are not jet defined. Classically you interpret them as a node that is connected to it self, but here this does not neccesary matter, since a connection from each node to it self, just couples the self interaction part of the update process with the neigbour interaction one, so in the learning phase, this difference will be made redundant. I will keep them to be one consistently in my examples, because i think that decompression algorithms look a bit nicer then.
+To work numerically with a graph, you can use the adjacency matrix of it, its entry #A_ij# are #1# if there is a connection between the nodes #i# and #j#<note this is a bit of a simplification, since the order of nodes does not matter, You can say that multiplication with any permutationmatrix does not neccesarily keep the adjacency matrix invariant, but still represents the same graph. I will try order my graphs from left to rigth or i`ll enumerate them> and #0# if not. You can extend this by replacing the #1# with a value defining the strength of a connection or by transforming the graph into a directed graph by removing the symmetry of #A# (#A_ij# is one exactly if there is a connection from #i# to #j#), but both of those extensions do not change much when we talk about decompression. Finally, there are the diagonal entries #A_ii# of the adjacency matrix, that are not jet defined. Classically you interpret them as a node that is connected to it self, but here this does not neccesary matter, since a connection from each node to it self, just couples the self interaction part of the update process with the neigbour interaction one, so in the learning phase, this difference will be made redundant. I will keep them to be one consistently in my examples, because i think that decompression algorithms look a bit nicer then.
 <i f="basic_graphs" wmode=True>a) a simple 3 node graph, b) a 2 node graph, c) a weigthed graph, d) a directed graph</i>
 
 <subsection what is a graph of a graph>
-To think about what it means, to have a graph of a graph, consider the following example: Lets say there is a room, with a lot of people inside. Each of them have one of three jobs: there are some fishermen, some doctors and a couple economists. In this group of people, each doctor likes to talk to anybody (and the other people are happy to do so), but economists and fishermen really dont like each other, while also everybody likes to talk to people with the same job<note so the diagonal elements of the adjacency matrix are one>. You can now use a graph to map those relations, and since we only have three kinds of people for now, this graph has three nodes (see the graph a below). Now lets introduce a second attribute for each person, maybe something like a popular movie, that the first half has seen and wants to talk about, which annoyes the second half. So you get a second graph (graph b below) with two kinds of people, who want to talk to each other, but never to the other group.
+To think about what it means, to have a graph of a graph, consider the following example: Lets say there is a room, with a lot of people inside. Each of them have one of three jobs: there are some fishermen, some doctors and a couple economists. In this group of people, each doctor likes to talk to anybody (and the other people are happy to do so), but economists and fishermen really don`t like each other, while also everybody likes to talk to people with the same job<note so the diagonal elements of the adjacency matrix are one>. You can now use a graph to map those relations, and since we only have three kinds of people for now, this graph has three nodes (see the graph a below). Now lets introduce a second attribute for each person, maybe something like a popular movie, that the first half has seen and wants to talk about, which annoyes the second half. So you get a second graph (graph b below) with two kinds of people, who want to talk to each other, but never to the other group.
 So we now have two graphs describing the situation if each person would only have one attribute, but can we describe the whole situation with a graph? and can we define a function to go from two partial adjacency matrices to a complete one? Yes (graph c), by just utilising the given rules: two people want to talk to each other, if both "attributes" want to talk to each other, or (using #n_1# as the number of nodes of the first graph)
 ##Eq(A(i_1+n_1*i_2,j_1+n_1*j_2),A(i_1,j_1)*A(i_2,j_2))##
 so basically each new connection is the logical and of both old connections
diff --git a/out/label.json b/out/label.json
index 79d7825c3d99c4501b0ef4997752d6e015a7c6f6..4f8a003a2fb9097fc93870dd095490eb58d786be 100644
--- a/out/label.json
+++ b/out/label.json
@@ -329,7 +329,7 @@
   },
   {
     "typ": "subsubsection",
-    "title": "Oneoff width",
+    "title": "oneoff width",
     "label": "evaloow",
     "file": "..\\..\\write\\/data\\03graphae\\07eval",
     "issec": true
@@ -454,7 +454,7 @@
   },
   {
     "typ": "section",
-    "title": "Problems",
+    "title": "Open questions",
     "label": "secinv",
     "file": "..\\..\\write\\/data\\04problems\\01intro\\01intro",
     "issec": true
@@ -623,7 +623,7 @@
   },
   {
     "typ": "section",
-    "title": "Solution 1: normalization",
+    "title": "Normalization",
     "label": "secnorm",
     "file": "..\\..\\write\\/data\\05normation\\00section",
     "issec": true
@@ -662,7 +662,7 @@
       "../imgs/aucmap1011"
     ],
     "label": "aucmap1011",
-    "caption": "(1011..maybe not the best)Aucmap for normally normated networks, showing nothing useful being learned",
+    "caption": "(1011..maybe not the best)Aucmap for normally normalized networks, showing nothing useful being learned",
     "where": "..\\..\\write\\/data\\05normation\\04norm"
   },
   {
@@ -681,7 +681,7 @@
       "../imgs/drtoptagging"
     ],
     "label": "drtoptagging",
-    "caption": "(generate later for computational reasons)double roc curve for invertibility of normated networks",
+    "caption": "(generate later for computational reasons)double roc curve for invertibility of normalized networks",
     "where": "..\\..\\write\\/data\\05normation\\04norm"
   },
   {
@@ -713,14 +713,14 @@
   },
   {
     "typ": "subsubsection",
-    "title": "Improving the AUC scores for normated networks",
+    "title": "Improving the AUC scores for normalized networks",
     "label": "normimpro",
     "file": "..\\..\\write\\/data\\05normation\\06aucs",
     "issec": true
   },
   {
     "typ": "subsubsection",
-    "title": "Scaling in normated networks",
+    "title": "Scaling in normalized networks",
     "label": "scalenorm",
     "file": "..\\..\\write\\/data\\05normation\\06aucs",
     "issec": true
@@ -731,7 +731,7 @@
       "../imgs/m4scaleroc"
     ],
     "label": "m4scaleroc",
-    "caption": "(multi4scale roc) AUC values for higher normated batches by their training data",
+    "caption": "(multi4scale roc) AUC values for higher normalized batches by their training data",
     "where": "..\\..\\write\\/data\\05normation\\06aucs"
   },
   {
@@ -765,19 +765,19 @@
       "../imgs/aucmap534"
     ],
     "label": "aucmap534",
-    "caption": "ABE good norm",
+    "caption": "AUC feature map for a well normated network",
     "where": "..\\..\\write\\/data\\05normation\\09normplus"
   },
   {
     "typ": "section",
-    "title": "Solution 2: Mixed networks",
+    "title": "Mixed networks",
     "label": "secmixed",
     "file": "..\\..\\write\\/data\\06othernets\\00intro",
     "issec": true
   },
   {
     "typ": "subsection",
-    "title": "Oneoff networks",
+    "title": "oneoff networks",
     "label": "oneoff",
     "file": "..\\..\\write\\/data\\06othernets\\02oneoff",
     "issec": true
@@ -804,12 +804,12 @@
       "../imgs/mabe3"
     ],
     "label": "mabe3",
-    "caption": "(multiabe 3)Auc as function of the epoch, trained on QCD, once for a graph oneOff and once for a dense oneOff. As you see, both relations show a maximum before the training ends, but the graph network is way more continuos",
+    "caption": "(multiabe 3)Auc as function of the epoch, trained on QCD, once for a graph oneoff and once for a dense oneoff. As you see, both relations show a maximum before the training ends, but the graph network is way more continuous",
     "where": "..\\..\\write\\/data\\06othernets\\02oneoff"
   },
   {
     "typ": "subsection",
-    "title": "Compressed oneOff learning",
+    "title": "Compressed oneoff learning",
     "label": "mixedidea",
     "file": "..\\..\\write\\/data\\06othernets\\03idea",
     "issec": true
@@ -823,14 +823,14 @@
   },
   {
     "typ": "subsubsection",
-    "title": "trained on QCD",
+    "title": "Trained on QCD",
     "label": "classQCD",
     "file": "..\\..\\write\\/data\\06othernets\\04areallygood",
     "issec": true
   },
   {
     "typ": "subsubsection",
-    "title": "trained on top",
+    "title": "Trained on top",
     "label": "classtop",
     "file": "..\\..\\write\\/data\\06othernets\\04areallygood",
     "issec": true
@@ -842,7 +842,7 @@
       "../imgs/seproc928"
     ],
     "label": "sephist928",
-    "caption": "oneoff loss distribution and roc curve for a network trained on top jets",
+    "caption": "Oneoff loss distribution and roc curve for a network trained on top jets",
     "where": "..\\..\\write\\/data\\06othernets\\04areallygood"
   },
   {
@@ -852,7 +852,7 @@
       "../imgs/ptdraw928"
     ],
     "label": "simpledraw928",
-    "caption": "Reconstruction images for a normated network trained on QCD",
+    "caption": "Reconstruction images for a normalized network trained on QCD",
     "where": "..\\..\\write\\/data\\06othernets\\04areallygood"
   },
   {
@@ -872,12 +872,12 @@
       "../imgs/ptdraw1128"
     ],
     "label": "simpledraw1128",
-    "caption": "Reconstruction images for a normated network trained on top",
+    "caption": "Reconstruction images for a normalized network trained on top",
     "where": "..\\..\\write\\/data\\06othernets\\04areallygood"
   },
   {
     "typ": "subsection",
-    "title": "Scale",
+    "title": "Scaling for oneoff networks",
     "label": "scale3",
     "file": "..\\..\\write\\/data\\06othernets\\06scale",
     "issec": true
@@ -903,7 +903,7 @@
   },
   {
     "typ": "section",
-    "title": "Other data",
+    "title": "Applying this model to other datasets",
     "label": "secdata",
     "file": "..\\..\\write\\/data\\07otherdata\\01intro",
     "issec": true
@@ -961,14 +961,14 @@
   },
   {
     "typ": "subsubsection",
-    "title": "Quark v gluon",
+    "title": "Quark or gluon",
     "label": "qg",
     "file": "..\\..\\write\\/data\\07otherdata\\03otherdata",
     "issec": true
   },
   {
     "typ": "subsubsection",
-    "title": "leptons",
+    "title": "Leptons",
     "label": "leptons",
     "file": "..\\..\\write\\/data\\07otherdata\\03otherdata",
     "issec": true
@@ -995,7 +995,7 @@
   },
   {
     "typ": "subsection",
-    "title": "Even more comparison",
+    "title": "Cross comparisons",
     "label": "crossdata",
     "file": "..\\..\\write\\/data\\07otherdata\\05cross",
     "issec": true
@@ -1363,7 +1363,7 @@
       "../imgs/none"
     ],
     "label": "none",
-    "caption": "(FORGOT THIS PLOT, AND CALCULATION TAKES TIME)A simple old AUC by epoch plot for a unnormated network with thus focus on angular data",
+    "caption": "(FORGOT THIS PLOT, AND CALCULATION TAKES TIME)A simple old AUC by epoch plot for a unnormalized network with thus focus on angular data",
     "where": "..\\..\\write\\/data\\10anhang\\04whyaesuck"
   },
   {
@@ -1478,7 +1478,7 @@
   },
   {
     "typ": "subsection",
-    "title": "Using oneOff networks on MNist data",
+    "title": "Using oneoff networks on MNist data",
     "label": "amnist",
     "file": "..\\..\\write\\/data\\10anhang\\15mnist.py",
     "issec": true
@@ -1648,26 +1648,26 @@
   },
   {
     "typ": "subsubsection",
-    "title": "Some physical interpretability for oneOff networks",
+    "title": "Some physical interpretability for oneoff networks",
     "label": "oometrik",
     "file": "..\\..\\write\\/data\\10anhang\\35oometrik",
     "issec": true
   },
   {
     "typ": "table",
-    "caption": "Learned metrik values of oneOff networks trained on muon events",
+    "caption": "Learned metrik values of oneoff networks trained on muon events",
     "label": "oneoffmuon",
     "file": "..\\..\\write\\/data\\10anhang\\35oometrik"
   },
   {
     "typ": "table",
-    "caption": "Learned metrik values for a diagonal metrik oneOff networks trained on muon events",
+    "caption": "Learned metrik values for a diagonal metrik oneoff networks trained on muon events",
     "label": "oneoffmuondiag",
     "file": "..\\..\\write\\/data\\10anhang\\35oometrik"
   },
   {
     "typ": "subsection",
-    "title": "How a oneOff network can become noninvertible",
+    "title": "How a oneoff network can become noninvertible",
     "label": "oofail",
     "file": "..\\..\\write\\/data\\10anhang\\36whyoneoffsfail",
     "issec": true
@@ -1679,7 +1679,7 @@
       "../imgs/dsantiinv25"
     ],
     "label": "drantiinv25",
-    "caption": "(doubleroc/sep)Antiinvertibility for ldm data, left using the autoencoder loss, and on rigth using oneOff networks",
+    "caption": "(doubleroc/sep)Antiinvertibility for ldm data, left using the autoencoder loss, and on rigth using oneoff networks",
     "where": "..\\..\\write\\/data\\10anhang\\36whyoneoffsfail"
   },
   {
@@ -1693,14 +1693,14 @@
   },
   {
     "typ": "subsection",
-    "title": "Treating oneOff networks as observables",
+    "title": "Treating oneoff networks as observables",
     "label": "observable",
     "file": "..\\..\\write\\/data\\10anhang\\37oopar",
     "issec": true
   },
   {
     "typ": "subsection",
-    "title": "Chancing the definition of the transverse momentum",
+    "title": "changing the definition of the transverse momentum",
     "label": "alpt",
     "file": "..\\..\\write\\/data\\10anhang\\38lpt",
     "issec": true
@@ -1889,7 +1889,7 @@
   },
   {
     "typ": "subsection",
-    "title": "Trainingsize, and why graph autoencoder dont care about it",
+    "title": "Trainingsize, and why graph autoencoder don`t care about it",
     "label": "asize",
     "file": "..\\..\\write\\/data\\10anhang\\85sizematters",
     "issec": true
diff --git a/out/main.aux b/out/main.aux
index 8d3b5ee6bf102a01e91bbc152d56ad4459850229..aba39053ae7391efaf8ff0a6572ab59c88aee902 100644
--- a/out/main.aux
+++ b/out/main.aux
@@ -47,11 +47,11 @@
 \citation{aeformotors}
 \citation{aefornuclear}
 \citation{graphbasic}
-\newlabel{fig:aemerge1_1}{{\caption@xref {fig:aemerge1_1}{ on input line 227}}{7}{Neuronal networks and autoencoder}{figure.caption.3}{}}
+\newlabel{fig:aemerge1_1}{{\caption@xref {fig:aemerge1_1}{ on input line 229}}{7}{Neuronal networks and autoencoder}{figure.caption.3}{}}
 \newlabel{sub@fig:aemerge1_1}{{}{7}{Neuronal networks and autoencoder}{figure.caption.3}{}}
-\newlabel{fig:aemerge1_2}{{\caption@xref {fig:aemerge1_2}{ on input line 232}}{7}{Neuronal networks and autoencoder}{figure.caption.3}{}}
+\newlabel{fig:aemerge1_2}{{\caption@xref {fig:aemerge1_2}{ on input line 234}}{7}{Neuronal networks and autoencoder}{figure.caption.3}{}}
 \newlabel{sub@fig:aemerge1_2}{{}{7}{Neuronal networks and autoencoder}{figure.caption.3}{}}
-\newlabel{fig:aemerge1_3}{{\caption@xref {fig:aemerge1_3}{ on input line 237}}{7}{Neuronal networks and autoencoder}{figure.caption.3}{}}
+\newlabel{fig:aemerge1_3}{{\caption@xref {fig:aemerge1_3}{ on input line 239}}{7}{Neuronal networks and autoencoder}{figure.caption.3}{}}
 \newlabel{sub@fig:aemerge1_3}{{}{7}{Neuronal networks and autoencoder}{figure.caption.3}{}}
 \@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces Example images showing how autoencoder can combine two images into one. Taken from \cite  {ostagram} (Ember, Bruno, ArgonOl)\relax }}{7}{figure.caption.3}\protected@file@percent }
 \newlabel{fig:aemerge1}{{1.2}{7}{Example images showing how autoencoder can combine two images into one. Taken from \cite {ostagram} (Ember, Bruno, ArgonOl)\relax }{figure.caption.3}{}}
@@ -144,17 +144,17 @@
 \newlabel{sec:evalloss}{{3.6.2}{24}{Losses}{subsubsection.3.6.2}{}}
 \@writefile{toc}{\contentsline {subsubsection}{\numberline {3.6.3}Images}{24}{subsubsection.3.6.3}\protected@file@percent }
 \newlabel{sec:evalimg}{{3.6.3}{24}{Images}{subsubsection.3.6.3}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.6.4}Oneoff width}{24}{subsubsection.3.6.4}\protected@file@percent }
-\newlabel{sec:evaloow}{{3.6.4}{24}{Oneoff width}{subsubsection.3.6.4}{}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.6.4}oneoff width}{24}{subsubsection.3.6.4}\protected@file@percent }
+\newlabel{sec:evaloow}{{3.6.4}{24}{oneoff width}{subsubsection.3.6.4}{}}
 \@writefile{toc}{\contentsline {subsection}{\numberline {3.7}Evaluating the autoencoder }{24}{subsection.3.7}\protected@file@percent }
 \newlabel{sec:evalae}{{3.7}{24}{Evaluating the autoencoder}{subsection.3.7}{}}
 \@writefile{toc}{\contentsline {subsubsection}{\numberline {3.7.1}4 nodes}{25}{subsubsection.3.7.1}\protected@file@percent }
 \newlabel{sec:ae4}{{3.7.1}{25}{4 nodes}{subsubsection.3.7.1}{}}
 \@writefile{lof}{\contentsline {figure}{\numberline {3.5}{\ignorespaces training history for a 4 node network\relax }}{25}{figure.caption.16}\protected@file@percent }
 \newlabel{fig:history979}{{3.5}{25}{training history for a 4 node network\relax }{figure.caption.16}{}}
-\newlabel{fig:simpledraw979_1}{{\caption@xref {fig:simpledraw979_1}{ on input line 671}}{25}{4 nodes}{figure.caption.17}{}}
+\newlabel{fig:simpledraw979_1}{{\caption@xref {fig:simpledraw979_1}{ on input line 673}}{25}{4 nodes}{figure.caption.17}{}}
 \newlabel{sub@fig:simpledraw979_1}{{}{25}{4 nodes}{figure.caption.17}{}}
-\newlabel{fig:simpledraw979_2}{{\caption@xref {fig:simpledraw979_2}{ on input line 676}}{25}{4 nodes}{figure.caption.17}{}}
+\newlabel{fig:simpledraw979_2}{{\caption@xref {fig:simpledraw979_2}{ on input line 678}}{25}{4 nodes}{figure.caption.17}{}}
 \newlabel{sub@fig:simpledraw979_2}{{}{25}{4 nodes}{figure.caption.17}{}}
 \@writefile{lof}{\contentsline {figure}{\numberline {3.6}{\ignorespaces Reconstruction images for a 4 node image. On the left for angles, and on the rigth for the momentum.\relax }}{25}{figure.caption.17}\protected@file@percent }
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@@ -162,9 +162,9 @@
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diff --git a/out/main.out b/out/main.out
index 84ee0e2c4d81491e93fc1bf065d36e00626ac3df..dfbe3d03402f6a5d52f33297fbd9a0673237b045 100644
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 \BOOKMARK [2][-]{subsection.9.11}{changing the input feature space }{section.9}% 105
 \BOOKMARK [2][-]{subsection.9.12}{Why c addition migth not be perfect }{section.9}% 106
-\BOOKMARK [2][-]{subsection.9.13}{Using oneOff networks on MNist data }{section.9}% 107
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 \BOOKMARK [2][-]{subsection.9.14}{Metrik analysis }{section.9}% 108
 \BOOKMARK [2][-]{subsection.9.15}{The compression algorithm that we would wish to have }{section.9}% 109
 \BOOKMARK [2][-]{subsection.9.16}{topK and better graphs }{section.9}% 110
@@ -113,10 +113,10 @@
 \BOOKMARK [3][-]{subsubsection.9.16.3}{oneoff math}{subsection.9.16}% 113
 \BOOKMARK [2][-]{subsection.9.17}{Self improving oneoff networks }{section.9}% 114
 \BOOKMARK [3][-]{subsubsection.9.17.1}{oneoff outside of physics}{subsection.9.17}% 115
-\BOOKMARK [3][-]{subsubsection.9.17.2}{Some physical interpretability for oneOff networks}{subsection.9.17}% 116
-\BOOKMARK [2][-]{subsection.9.18}{How a oneOff network can become noninvertible }{section.9}% 117
-\BOOKMARK [2][-]{subsection.9.19}{Treating oneOff networks as observables }{section.9}% 118
-\BOOKMARK [2][-]{subsection.9.20}{Chancing the definition of the transverse momentum }{section.9}% 119
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+\BOOKMARK [2][-]{subsection.9.18}{How a oneoff network can become noninvertible }{section.9}% 117
+\BOOKMARK [2][-]{subsection.9.19}{Treating oneoff networks as observables }{section.9}% 118
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 \BOOKMARK [2][-]{subsection.9.22}{Comparing our graph update layer to particleNet }{section.9}% 121
 \BOOKMARK [2][-]{subsection.9.23}{Variating the compression size }{section.9}% 122
@@ -130,7 +130,7 @@
 \BOOKMARK [2][-]{subsection.9.28}{The usage of a batchNormalization layer in the middle of the gae }{section.9}% 130
 \BOOKMARK [2][-]{subsection.9.29}{Uncertainity and reproducability of AUC values }{section.9}% 131
 \BOOKMARK [2][-]{subsection.9.30}{Is it a good idea to relearn the graph at each step }{section.9}% 132
-\BOOKMARK [2][-]{subsection.9.31}{Trainingsize, and why graph autoencoder dont care about it }{section.9}% 133
+\BOOKMARK [2][-]{subsection.9.31}{Trainingsize, and why graph autoencoder don`t care about it }{section.9}% 133
 \BOOKMARK [2][-]{subsection.9.32}{Graph autoencoder as autoencoder with some graph layers in front }{section.9}% 134
 \BOOKMARK [2][-]{subsection.9.33}{Why autoencoder reproduce mean values }{section.9}% 135
 \BOOKMARK [1][-]{section.10}{appendix 2: Other usecases}{}% 136
diff --git a/out/main.pdf b/out/main.pdf
index 9d8391d722da7e33dd0f8e7794438e3e21362720..a2d211ae7cf682c1c2f100650e3800fa9497b03a 100644
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 %%EOF
diff --git a/out/main.tex b/out/main.tex
index 152cee39e7e142b91c4a4dfb7a65f3db6780768b..0a12a54aedcd7e602ad67693002f080ed0c5f33d 100644
--- a/out/main.tex
+++ b/out/main.tex
@@ -60,7 +60,7 @@
 	\centering
 	{\huge\bfseries Deep learning for new physics mining at the LHC\par}
 	\vspace{2cm}
-	{\LARGE\itshape Simon kluettermann\par}
+	{\LARGE\itshape Simon Kluettermann\par}
 	\vspace{1.5cm}
 	{\scshape\Large Master Thesis in Physics\par}
 	\vspace{0.2cm}
@@ -96,6 +96,7 @@
 \pagenumbering{arabic}
 
 
+
 %from file ..\..\write\/data\00basic\01toc
 \renewcommand*\contentsname{Table of content}
 \tableofcontents
@@ -113,6 +114,7 @@
 \section{Introduction and literature}\label{sec:secintro}
 
 
+
 %from file ..\..\write\/data\01intro\01intro
 \subsection{Motivation }\label{sec:intro}
 
@@ -144,7 +146,7 @@ Current state of this thesis
 
       \item some results not yet final(just numbers i found somewhere, not yet optimized, so things look a bit worse than they hopefully will)
 
-      \item spellchecking not done. I wrote a programm for this, but after half an hour it bugged out, and i dont have the patience to fix it now. So there are spelling errors, but you can just ignore them
+      \item spellchecking not done. I wrote a programm for this, but after half an hour it bugged out, and i don`t have the patience to fix it now. So there are spelling errors, but you can just ignore them
 
       \item all references missing...those i do first thing tomorrow
 
@@ -190,7 +192,7 @@ Current state of this thesis
 \subsection{New physics }\label{sec:physics}
 
 
-Modern particle physics seems to be in a standstill: The standart model seems to explain everything on a small size, while also beeing clearly incomplete. At the same time, each suggested extension seems to be either wrong or untestable, which is why there are now approaches changing the fundamental way we do science. One of this approaches is suggested by QCDorWhat \cite{QCDorWhat}: Instead of finding new physics events by hypotising theories and testing them afterwards, use an anomaly detection algorithm to filter out events that dont match your expectation. This allows you to find events representing new physics models without needing to suggest these models first. They work here in jet physics, trying to find anomalous jets that are generated by the decay of a top quark, while only knowing about those jets, which are generated by the decay of QCD\footnote{Quantum chromodynamics, with QCD particles you usually mean low mass quarks and gluons.} particles with lower mass. You can think of this task, as trying to finding new physics, while only knowing as much as we did before the detection of the top quark in 1995\cite{topquark}. This suggests that when we would be able to find top quarks at this point in time, we migth also be able to apply such an algorithm to the LHC now and use it to understand physics no human knows about yet.
+Modern particle physics seems to be in a standstill: The standart model seems to explain everything on a small size, while also beeing clearly incomplete. At the same time, each suggested extension seems to be either wrong or untestable, which is why there are now approaches changing the fundamental way we do science. One of this approaches is suggested by QCDorWhat \cite{QCDorWhat}: Instead of finding new physics events by hypotising theories and testing them afterwards, use an anomaly detection algorithm to filter out events that don`t match your expectation. This allows you to find events representing new physics models without needing to suggest these models first. They work here in jet physics, trying to find anomalous jets that are generated by the decay of a top quark, while only knowing about those jets, which are generated by the decay of QCD\footnote{Quantum chromodynamics, with QCD particles you usually mean low mass quarks and gluons.} particles with lower mass. You can think of this task, as trying to finding new physics, while only knowing as much as we did before the detection of the top quark in 1995\cite{topquark}. This suggests that when we would be able to find top quarks at this point in time, we migth also be able to apply such an algorithm to the LHC now and use it to understand physics no human knows about yet.
 
 
 
@@ -219,7 +221,7 @@ Autoencoder \cite{aebasic} are a special kind of these neural networks, in which
 As seen in \ref{fig:aeexamp}, to completely encode the data you would still require 2 dimensions (a $x$ and a $y$ value), but you can approximately encode them into 1 dimension quite well, by using one value as this compressed state and reconstruct the second one in the decoder, as a linear function of the compressed state\footnote{Since the number of trainingsamples is finite, you could map every sample into an index, and map those indices again onto the inputs. This would reach a zero loss for any input with an compression size of 1. The problem is that not only is finding such a function quite hard for a neural network, it would also not be useful at all, since on any new data (for example the validation data), the network would not work at all. This is why these kind of functions are a part of what you can call overfitting for autoencoder.}\footnote{In practice, this data contains structural noise, which is why the autoencoder would not learn a linear function, but a more complicated on, better representing the data.}.
 
 
-This combination of a compresser and a decompresser can be quite useful in multiple ways. Ignoring the obvious task of compressing data (see \cite{aeforcompress} for an example of an autoencoder used in particle physics), you can give the decompressor noise to generate new versions of an already known kind of data (see \cite{aeforgen}), and even though nowadays GANs(Generative adversial networks see \cite{aeforgen} or \ref{sec:tipsy}) are used for this task, autoencoder have still some benefits, allowing for more control over the generated data. This works, since (for good autoencoders\footnote{This works better in a special way of training an autoencoder, called a variational autoencoder \cite{variationalae}.}) similarity in the compressed space represent similarity of the inputs. This does mean, that by identifiyng features in the input space you can chance just one attribute of an input, and you can also combine the features of two inputs into one, see for example \cite{aecombine}
+This combination of a compresser and a decompresser can be quite useful in multiple ways. Ignoring the obvious task of compressing data (see \cite{aeforcompress} for an example of an autoencoder used in particle physics), you can give the decompressor noise to generate new versions of an already known kind of data (see \cite{aeforgen}), and even though nowadays GANs(Generative adversial networks see \cite{aeforgen} or \ref{sec:tipsy}) are used for this task, autoencoder have still some benefits, allowing for more control over the generated data. This works, since (for good autoencoders\footnote{This works better in a special way of training an autoencoder, called a variational autoencoder \cite{variationalae}.}) similarity in the compressed space represent similarity of the inputs. This does mean, that by identifiyng features in the input space you can change just one attribute of an input, and you can also combine the features of two inputs into one, see for example \cite{aecombine}
 \begin{figure}[H]
 \begin{subfigure}{0.3\textwidth}
   \centering
@@ -276,7 +278,7 @@ A graph\cite{graphbasic} is a mathematical concept, that allows to define a more
   \end{figure}
 
 
-Mathematically, these nodes and relations are defined in a list of feature vectors\footnote{Technically this is equivalent to a matrix, but list of vectors is more intuitive.} $X$ that stores the features of each node, and an adjacency matrix $A$, which components $A_{i}^{j}$ are $1$, if the nodes $i$ and $j$ are connected, and $0$ if not. This graph is usually invariant under permuation of the node indices. You achieve this by permuting\footnote{With permuting we here mean switching two indices, or more generally multiplying with a permutation matrix.} the adjacency matrix in the same way the feature vectors are permuted\footnote{This is the reason why we dont call the list of feature vectors a matrix: as a matrix permutation requires permutation matrices on each side ($A \cdot p \cdot p$), the feature vector "Matrix" only requires one permutation matrix ($X \cdot p$).}, and by requiring any action on the graph to be permuation invariant. This action is usually also local, and thus only acts on each node and the mean\footnote{You also need to require each local action to be symmetric under chancing the input ordering, since else the output can depend on the order of the nodes. The usual way that is achieved is by using a function like the mean on all neighbours.} of nodes that are connected to the current node\footnote{This works, since permuting indices does not chance, which nodes are connected together.}. This has the benefit of making graphs ideal for modeling interactions between high numbers of objects, as the functions dont chance as you add more nodes to the graph. In informatics, this is useful for example for social networks\cite{graphsoc}: Data that consists out of a huge amount of nodes in which mostly only connected nodes (friends) affect each other, are perfect applications for graphs, since else you would need to update your model every time a new user joins. In physics, this reminds of nuclear science, and the approximation of pair interaction potentials\cite{pairpot}, and so there are applications using this kind of molecule encoding for chemical feature extraction (for a simple example look at chapter \ref{sec:mol}) \cite{gnnforchemistry} and medicine \cite{gnnformedicine}.
+Mathematically, these nodes and relations are defined in a list of feature vectors\footnote{Technically this is equivalent to a matrix, but list of vectors is more intuitive.} $X$ that stores the features of each node, and an adjacency matrix $A$, which components $A_{i}^{j}$ are $1$, if the nodes $i$ and $j$ are connected, and $0$ if not. This graph is usually invariant under permuation of the node indices. You achieve this by permuting\footnote{With permuting we here mean switching two indices, or more generally multiplying with a permutation matrix.} the adjacency matrix in the same way the feature vectors are permuted\footnote{This is the reason why we don`t call the list of feature vectors a matrix: as a matrix permutation requires permutation matrices on each side ($A \cdot p \cdot p$), the feature vector "Matrix" only requires one permutation matrix ($X \cdot p$).}, and by requiring any action on the graph to be permuation invariant. This action is usually also local, and thus only acts on each node and the mean\footnote{You also need to require each local action to be symmetric under changing the input ordering, since else the output can depend on the order of the nodes. The usual way that is achieved is by using a function like the mean on all neighbours.} of nodes that are connected to the current node\footnote{This works, since permuting indices does not change, which nodes are connected together.}. This has the benefit of making graphs ideal for modeling interactions between high numbers of objects, as the functions don`t change as you add more nodes to the graph. In informatics, this is useful for example for social networks\cite{graphsoc}: Data that consists out of a huge amount of nodes in which mostly only connected nodes (friends) affect each other, are perfect applications for graphs, since else you would need to update your model every time a new user joins. In physics, this reminds of nuclear science, and the approximation of pair interaction potentials\cite{pairpot}, and so there are applications using this kind of molecule encoding for chemical feature extraction (for a simple example look at chapter \ref{sec:mol}) \cite{gnnforchemistry} and medicine \cite{gnnformedicine}.
 Next to those relational applications, there are also applications that are not utilizing an existing relation, but use the locality of the graph structure to encode the similarity of given data. This is done by letting the sense of similarity between nodes be a learnable function. For example, by using a topK algorithm (each node is connected to its neirest $K$ neighbours, see \ref{sec:atopk}), you can implement a learnable version of whatever distance means. This allows networks like for example particleNet \cite{particeNet}, which uses a special kind of neural network, that is able to work on graphs, to seperate top and QCD jets\footnote{With QCD we mean jets that are generated by gluons or other quarks that are not top quarks.} in a supervised way. They use the graph structure, to be able to define and redefine multiple times, which detected particles (nodes), should be considered close to each other. This results in particleNet beeing a quite good classifier(see \cite{toptagref}).
 
 
@@ -378,7 +380,7 @@ Or to focus on the fraction of falsy called signal events, which is usually call
 
 \subsubsection{Area under the curve}\label{sec:classauc}
 {\scriptsize Referenced in: [\ref{sec:classauc}] [\ref{sec:evaloow}]  \par}
-To simplify comparing ROC scores, you can use an AUC (Area under the curve) score to summarise it. This AUC score is defined as the integral of the true positive rate over the false positive rate. This simplification is not perfect, since you reduce a function into only one number, but it is fairly wide accepted, as it is much easier to interpret: A perfect score would result in an AUC score of 1, while a classifier that just guesses randomly, results in an AUC score of 0.5 and a perfect anticlassifier would result in 0.  On the other hand, since not every part of the roc curve is equally important for the current problem (if you want to test, if somebody is ill, you migth prefer more false positives over more hidden illnesses). This could result in networks improving the AUC score by just chancing unimportant parts of the ROC curve.
+To simplify comparing ROC scores, you can use an AUC (Area under the curve) score to summarise it. This AUC score is defined as the integral of the true positive rate over the false positive rate. This simplification is not perfect, since you reduce a function into only one number, but it is fairly wide accepted, as it is much easier to interpret: A perfect score would result in an AUC score of 1, while a classifier that just guesses randomly, results in an AUC score of 0.5 and a perfect anticlassifier would result in 0.  On the other hand, since not every part of the roc curve is equally important for the current problem (if you want to test, if somebody is ill, you migth prefer more false positives over more hidden illnesses). This could result in networks improving the AUC score by just changing unimportant parts of the ROC curve.
 
 
 
@@ -405,12 +407,12 @@ Every network has a fixed maximum number of particles that can be put into it, f
     \item $\Delta_{\phi}$: $\phi = \operatorname{arctan_{2}}{\left(p_{2},p_{1} \right)}$\footnote{The function $\operatorname{atan_{2}}{\left(y,x \right)}$ is an extension of $\arctan{\left(y / x \right)}$ that is able to map to the full $2 \cdot \pi$ output space.} which is again shiftet in such a way, that the mean of $\Delta_{\phi}$ is 0: $\Delta_{\phi} = \phi - \operatorname{mean}{\left(\phi \right)}$\footnote{Here is shifting actually not that easy to implement, since you have to consider the difference in a modular space, see appendix \ref{sec:aimplementationphi} or the implementation (ENTER GIT LINK gpre5) for more information.}, since also this position of the jet should not have any meaning.
 
 
-    \item $lp_{T}$: $p_{T}^{2} = p_{1}^{2} + p_{2}^{2}$, and $lp_{T} = - \log{\left(\frac{p_{T}}{p_{T}^{jet}} \right)}$. This logarithm, is needed to keep each value of about the same order of magnitude, which makes the training more stable. We also divide by the total jet transverse momentum, to make every jet look more similar (see appendix \ref{sec:alpt} for the effects of chancing this). Finally the sign is used to keep the values positive. \footnote{A consequence is that higher transverse momenta have lower values. Since the alternative in appendix \ref{sec:alpt} changes this, we can say that this does not matter to much.}
+    \item $lp_{T}$: $p_{T}^{2} = p_{1}^{2} + p_{2}^{2}$, and $lp_{T} = - \log{\left(\frac{p_{T}}{p_{T}^{jet}} \right)}$. This logarithm, is needed to keep each value of about the same order of magnitude, which makes the training more stable. We also divide by the total jet transverse momentum, to make every jet look more similar (see appendix \ref{sec:alpt} for the effects of changing this). Finally the sign is used to keep the values positive. \footnote{A consequence is that higher transverse momenta have lower values. Since the alternative in appendix \ref{sec:alpt} changes this, we can say that this does not matter to much.}
 
 
 \end{itemize}
 
-You could try to use more variables: ParticleNet for example uses 4 more variables (different representations of other variables and the energy as well as $\Delta_{R}^{2} = \Delta_{H}^{2} + \Delta_{\Phi}^{2}$. The also dont use flag as input), but since these variables are strongly related to other variables, this results in an autoencoder only learning those relation. This would not result in anything learned beeing usable as classifier. And demanding that this and more is learned, just complicates the task, without providing any real benefit\footnote{But see appendix \ref{sec:afeature} for some experiments in chancing the features.}.
+You could try to use more variables: ParticleNet for example uses 4 more variables (different representations of other variables and the energy as well as $\Delta_{R}^{2} = \Delta_{H}^{2} + \Delta_{\Phi}^{2}$. The also don`t use flag as input), but since these variables are strongly related to other variables, this results in an autoencoder only learning those relation. This would not result in anything learned beeing usable as classifier. And demanding that this and more is learned, just complicates the task, without providing any real benefit\footnote{But see appendix \ref{sec:afeature} for some experiments in changing the features.}.
 
 
 
@@ -480,7 +482,7 @@ Our graph update layer consists out of two matrices, a self interaction matrix,
 So written as a formula, the new vector equals (with the original feature vector $x_{i}$, the learnable self and neigbour matrices $s_{i}^{j}$ and $n_{i}^{j}$, as well as the adjacency matrix $A_{i}^{j}$ and the activation $f$)
 \begin{equation}f{\left(A_{k}^{i} \cdot n_{j}^{k} \cdot x_{i} + s_{j}^{i} \cdot x_{i} \right)}\end{equation}
 (FORMULA GETS REORDERED LATER!)
-It should be noted, that this implementation has one central problem: It is a bit slower than the usual approach(for a reasoning on why we cannot use the more usual approach of particleNet, see appendix \ref{sec:acomparepnet})\footnote{Especcially since they can utilise GPUs better.}, and even though we dont think the implementation (see git (ENTER LINK)) is as fast possible, this is something that could be improved a lot .
+It should be noted, that this implementation has one central problem: It is a bit slower than the usual approach(for a reasoning on why we cannot use the more usual approach of particleNet, see appendix \ref{sec:acomparepnet})\footnote{Especcially since they can utilise GPUs better.}, and even though we don`t think the implementation (see git (ENTER LINK)) is as fast possible, this is something that could be improved a lot .
 
 
 \subsubsection{Tensorproducts}\label{sec:tensorproduct}
@@ -622,11 +624,11 @@ We migth be able to evaluate a binary classification problem\footnote{See chapte
 
 \subsubsection{AUC scores}\label{sec:evalauc}
 
-If you want to evaluate a network, you migth simply use the quality of the classifier (the AUC Score, see chapter \ref{sec:classauc}): since the classifier should work by the autoencoder understanding the data, and thus should only be good if also the autoencoder is good. And in most cases this works, there is a clear relation between the quality of the autoencoder and the quality of the classifier (see chapter \ref{sec:normalization}), but in general this is simply not true, as for example chapter \ref{sec:simplicity} show. And even if your working in a region where this relation is true, Classifier evaluation methods\footnote{AUC scores even have one of the lower uncertainities.} usually have a much higher uncertainity\footnote{Uncertainity in the sense that even a well trained network can chance its AUC score by a couple of percent after retraining, even if it has the same loss (see chapter \ref{sec:auncertain}).} than other methods, which is why in the regions in which there is a strong correlation, it was more useful to use the loss of the network to assert that the network improves, and to simply know that the AUC score will correlate.
+If you want to evaluate a network, you migth simply use the quality of the classifier (the AUC Score, see chapter \ref{sec:classauc}): since the classifier should work by the autoencoder understanding the data, and thus should only be good if also the autoencoder is good. And in most cases this works, there is a clear relation between the quality of the autoencoder and the quality of the classifier (see chapter \ref{sec:normalization}), but in general this is simply not true, as for example chapter \ref{sec:simplicity} show. And even if your working in a region where this relation is true, Classifier evaluation methods\footnote{AUC scores even have one of the lower uncertainities.} usually have a much higher uncertainity\footnote{Uncertainity in the sense that even a well trained network can change its AUC score by a couple of percent after retraining, even if it has the same loss (see chapter \ref{sec:auncertain}).} than other methods, which is why in the regions in which there is a strong correlation, it was more useful to use the loss of the network to assert that the network improves, and to simply know that the AUC score will correlate.
 
 \subsubsection{Losses}\label{sec:evalloss}
 
- Using only the quality of your autoencoder and trying to optimize this would be conceptually great, as you only need to use your anomalous data once\footnote{Usual machine learning has a problem, in which you network can learn even data that it is not trained on, simply by you comparing networks on it (this is why there is test data), the same can happen here, by you often comparing qualities of your anomalous data and since finding new test data would require you to have completely different anomalous systems, this can be difficult to do (even though we try this in chapter \ref{sec:secdata}), which is why choosing to ignore your anomalies in training would be great.}, but this again has problems: Not only requires this still a strong relation between AUC and loss (That is here given even less, consider the problem of finding the best compression size: The loss will usually\footnote{Always, except for noise and random chance.} fall by increasing the compression size, but at some point, the autoencoder can just reconstruct everything perfectly, and thus has no more classification potential), but the loss also relies heavily on the definition of the network and the normalization of the input data\footnote{See chapter \ref{sec:data}.}, which makes comparing different networks only possible, if you neither alter the loss nor the normalization.
+ Using only the quality of your autoencoder and trying to optimize this would be conceptually great, as you only need to use your anomalous data once\footnote{Usual machine learning has a problem, in which you network can learn even data that it is not trained on, simply by you comparing networks on it (this is why there is test data), the same can happen here, by you often comparing qualities of your anomalous data and since finding new test data would require you to have completely different anomalous systems, this can be difficult to do (even though we try this in chapter \ref{sec:secdata}), which is why choosing to ignore your anomalies in training would be great.}, but this again has problems: Not only requires this still a strong relation between AUC and loss (That is here given even less, consider the problem of finding the best compression size: The loss will usually\footnote{Always, except for noise and random change.} fall by increasing the compression size, but at some point, the autoencoder can just reconstruct everything perfectly, and thus has no more classification potential), but the loss also relies heavily on the definition of the network and the normalization of the input data\footnote{See chapter \ref{sec:data}.}, which makes comparing different networks only possible, if you neither alter the loss nor the normalization.
 
 \subsubsection{Images}\label{sec:evalimg}
 
@@ -634,10 +636,10 @@ This cross comparison problem can be easily solved by simply looking at the imag
 
 This you can solve, by also looking at the $lp_{T}$ reproduction, but this demands weighting importance between images, and thus does not make evaluating images any easier.
 
-\subsubsection{Oneoff width}\label{sec:evaloow}
+\subsubsection{oneoff width}\label{sec:evaloow}
 
 
-The final solution, and the solution that seems to be the best currently, is based on the things introduced in chapter \ref{sec:secmixed}. Because of this, it will be explained in chapter \ref{sec:oneoff}. It works, by defining the loss in a way, that does not chance by chancing the loss function or the initial normalization. We do this by letting the network define its own variable, which should be constant over all background events. And by setting this constant to be $1$, the variance of this measurable is independent under chancing the inputs. This still requires some correlation between the variance of this variable and the AUC score, which we cannot assume in general, but chapter \ref{sec:impro} at least suggests that this is common.
+The final solution, and the solution that seems to be the best currently, is based on the things introduced in chapter \ref{sec:secmixed}. Because of this, it will be explained in chapter \ref{sec:oneoff}. It works, by defining the loss in a way, that does not change by changing the loss function or the initial normalization. We do this by letting the network define its own variable, which should be constant over all background events. And by setting this constant to be $1$, the variance of this measurable is independent under changing the inputs. This still requires some correlation between the variance of this variable and the AUC score, which we cannot assume in general, but chapter \ref{sec:impro} at least suggests that this is common.
 
 
 
@@ -732,7 +734,7 @@ and thus there reconstruction is also worse
 
 
 
-As you see in image \ref{fig:roc4}, these 4 particle networks already reach an AUC score of over $0.81$, which is quite good considering we only use 4 particles. By chancing this networks parameters you can reach AUCs upwards of $0.85$.
+As you see in image \ref{fig:roc4}, these 4 particle networks already reach an AUC score of over $0.81$, which is quite good considering we only use 4 particles. By changing this networks parameters you can reach AUCs upwards of $0.85$.
 
 \begin{figure}[H]
 \begin{subfigure}{0.45\textwidth}
@@ -809,7 +811,7 @@ Again, the most important feature are the angles, they alone would reach an auc
 
 %from file ..\..\write\/data\04problems\01intro\01intro
 \newpage
-\section{Problems}\label{sec:secinv}
+\section{Open questions}\label{sec:secinv}
 {\scriptsize Referenced in: [\ref{sec:mixedidea}] [\ref{sec:crossdata}] [\ref{sec:quickres1}] [\ref{sec:decoding}] [\ref{sec:nogaegeneral}]  \par}
 
 Given the results of the classifier introduced in the last chapter, there are two problems that limit the usefulness of these autoencoder. This chapter tries to understand them further, so that chapter \ref{sec:secnorm} and \ref{sec:secmixed} can solve them.
@@ -999,9 +1001,9 @@ as you see, for a low number of particles, this works fairly well, but then, at
 
 This better classifier reaches an AUC of over $0.915$ that is comparable to the best anomaly detection networks, for example QCDorWhat\cite{QCDorWhat} reaches $0.93$\footnote{$0.9255$ when you calculate the AUC from their roc curve.} but on sligthly different data, while the work of thorben finke reached $0.908$ on the same data\cite{thorb}.  So what is the value of those complicated models, if they only improve the AUC by at most single percentage differences.
 That beeing said, you also cannot assume that new physics has the same angular distribution difference, as QCD compared to top, making this alternative model useless in the task of finding new physics\footnote{Or at least useless unless you search for one specific kind of abnormal data, there are some examples showing other kinds of abnormal data behaving completely differently in \ref{sec:secdata}.},so the question of interrest is just: do complicated models contain something more than this trivial difference? And unfortunately this is very hard to test. C addition allows you to estimate the effect any small additional AUC would have, and an AUC of about $0.6$, optimally combined would only improve an AUC score of $0.9$ to $0.904$, while $0.7$ would improve up to $0.917$, so both improvements would probably be neirly unmeasurable. So there migth be some hidden effect in a hidden model, that allows them to find new physics\footnote{Please note, that we assume here a lot: first, that in $p_{T}$ there is potential to differentiate all kinds of new physics, that this potential is used perfectly by an algorithm that did not work as expected for angles, and also, maybe most improbable, that the loss is used perfectly, and there is no confusion from the angular part at all.}.
-But what we can say, is that the networks we looked at so far, probably dont do anything more than looking at angular information\footnote{Since the classifier improves when you remove momenta, and the resulting classifier again improves when you replace the returned angles by zero.} and thus their fairly good AUC score is completely useless for finding new physics. What we want to do in the following is to force our network to learn something nontrivial, and thus actually create a correlation between how the network works on top jet anomalies and how it would behave on new physics.
+But what we can say, is that the networks we looked at so far, probably don`t do anything more than looking at angular information\footnote{Since the classifier improves when you remove momenta, and the resulting classifier again improves when you replace the returned angles by zero.} and thus their fairly good AUC score is completely useless for finding new physics. What we want to do in the following is to force our network to learn something nontrivial, and thus actually create a correlation between how the network works on top jet anomalies and how it would behave on new physics.
 Probably the most important result of this trivial model, is the effect it has on how to evaluate a model. We already talked about why just choosing a model that has a good AUC is a bad idea in chapter \ref{sec:evalprob}, but here this could probably not be clearer:
-Since we tried to train a model to be a great seperator, when we chanced the initial normalization, we choose those, that makes the network ignore angles and focus on $p_{T}$. This seamed useful, since this makes the model more like the trivial one and thus get a good AUC. But this also means, that we dont have a good autoencoder anymore, since a worse reconstruction can actually improve the quality of the network.
+Since we tried to train a model to be a great seperator, when we changed the initial normalization, we choose those, that makes the network ignore angles and focus on $p_{T}$. This seamed useful, since this makes the model more like the trivial one and thus get a good AUC. But this also means, that we don`t have a good autoencoder anymore, since a worse reconstruction can actually improve the quality of the network.
 You can also see this in the number of nodes we use: 4 node networks seemed to result in good classifiers, and you can clearly see a pattern in the first 4 nodes in images \ref{fig:tscale} and \ref{fig:tscaleno}.
 
 
@@ -1030,7 +1032,7 @@ All of this is obviously quite problematic, which is why the next two chapters (
 
 %from file ..\..\write\/data\05normation\00section
 \newpage
-\section{Solution 1: normalization}\label{sec:secnorm}
+\section{Normalization}\label{sec:secnorm}
 {\scriptsize Referenced in: [\ref{sec:secinv}] [\ref{sec:invertibility}]  \par}
 
 
@@ -1054,19 +1056,19 @@ One thing, We did not realise before trying to normalise the input datapoints, i
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/aucmap1011}
-\caption{(1011..maybe not the best)Aucmap for normally normated networks, showing nothing useful being learned}
+\caption{(1011..maybe not the best)Aucmap for normally normalized networks, showing nothing useful being learned}
 \label{fig:aucmap1011}
   \end{figure}
 
 
-This migth seem now, as if there is a trivial solution: just reduce the compression size accordingly, but this has three problems
+This seems as if there is a trivial solution: just reduce the compression size accordingly, but this has three problems
 \begin{itemize}
 
-    \item First, it is not completely trivial to misuse the normalization (Think of the standart deviation, there is a formula giving you information about the 4th value, given the first three. But even if we ignore the mean as beeing 0, this formula still involves squares and roots, which the network has to learn, and even then, there are always two possibilities for the resulting value.). So assuming that this is trivial, and that the network will always learn it garantueed, would be wrong
+    \item First, it is not completely trivial to misuse the normalization (Think of the standart deviation, there is a formula giving you information about the 4th value, given the first three. But even if we ignore the mean as beeing 0, this formula still involves squares and roots, which the network has to learn, and even then, there are always two possibilities for the resulting value.). So assuming that this is trivial, and that the network will always learn it garantueed, would be wrong.
 
-    \item Even if this is learned, this would not be enough: the network still has to compress this information further and this can lead to situations in which the network has to decide between learning the easy compression and the learning the interresting compression. In this situations it will probably always learn the trivial one
+    \item Even if this is learned, this would not be enough: the network still has to compress this information further and this can lead to situations in which the network has to decide between learning the easy compression and the learning the interresting compression. In this situations it will probably always learn the trivial one.
 
-    \item Trying to compress data with removed information further is not as easy as compressing non removed information. Think of two values distributed between $0$ and $10$: For example $4$ and $6$, or, after setting the mean to be $0$: $-1$ and $1$. In $4$ and $6$ the network can still decide to just average both values and get an mediocre prediction of $5$, which still describes these values in a way. But if after removing the mean, the network still averages both values, the predicted $0$ is fairly useless\footnote{Please note, that the difference between a good normalization and a bad one is just physical intuition. For example, we still set the mean of the jet angles to be zero, just because the direction relative to the measurement should be unimportant.}. From this we conclude, that simply subtracting each fixed value from the compression size does not work, as we would expect less good classifier.
+    \item Trying to compress data with removed information further is not as easy as compressing non removed information. Think of two values distributed between $0$ and $10$: For example $4$ and $6$, or, after setting the mean to be $0$: $-1$ and $1$. In $4$ and $6$ the network can still decide to just average both values and get an mediocre prediction of $5$, which still describes these values in a way. But if after removing the mean, the network still averages both values, the predicted $0$ is fairly unproductive\footnote{Note, that the difference between a good normalization and a bad one is just physical intuition. For example, we still set the mean of the jet angles to be zero, just because the direction relative to the measurement should be unimportant.}. From this we conclude, that simply subtracting each fixed value from the compression size does not work, as we would expect less good classifier.
 
 
 \end{itemize}
@@ -1078,12 +1080,12 @@ So what we need, is a better way of normalising the input data. From our thougth
 
     \item Scale invariance:$n{\left(x \right)} = n{\left(a \cdot x \right)}$
 
-    \item No fixed features: you can not write any $f{\left(x \right)}$ so that $f{\left(n{\left(x \right)} \right)} = 0$ for every $x$ is given
+    \item No fixed features: you can not write any $f{\left(x \right)}$ so that $f{\left(n{\left(x \right)} \right)} = 0$ for every $x$ is given.
 
 
 \end{itemize}
 
-The first two rules are obvious, since we want to use this, to remove any size information, and third rule would solve the problem of an autoencoder focussing on normalization artefacts\footnote{This last rule would actually be solved by demanding the standart deviation to be constant.}.
+The first two rules are obvious, since we want to use this, to remove any size information, and the third rule would solve the problem of an autoencoder focussing on normalization artefacts\footnote{This last rule would actually be solved by demanding the standart deviation to be constant.}.
 
 All three rules\footnote{Except for scale invariance with $a \leq 0$.} are solved by the following 3 normalization steps ($x$ is the input, $n$ the output of the normalization method)
 
@@ -1120,7 +1122,7 @@ But the quality suffers
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/drtoptagging}
-\caption{(generate later for computational reasons)double roc curve for invertibility of normated networks}
+\caption{(generate later for computational reasons)double roc curve for invertibility of normalized networks}
 \label{fig:drtoptagging}
   \end{figure}
 
@@ -1136,7 +1138,7 @@ Also networks, that before were very reproducable in their training\footnote{Whi
   \end{figure}
 
 
-This relation is great, since it means, that finding a better autoencoder, automatically results in a better classifier, and we thus can focus completely on improving the autoencoder. 
+This relation is very useful, since it means, that finding a better autoencoder, automatically results in a better classifier, and we thus can focus completely on improving the autoencoder. 
 Also by looking at this relation, we are able to justify the new compression size
 \begin{figure}[H] 
   \centering
@@ -1168,20 +1170,20 @@ training for 1000 epochs, and then until the loss increases for 250 epochs, resu
 
 
 %from file ..\..\write\/data\05normation\06aucs
-\subsubsection{Improving the AUC scores for normated networks}\label{sec:normimpro}
+\subsubsection{Improving the AUC scores for normalized networks}\label{sec:normimpro}
 {\scriptsize Referenced in: [\ref{sec:scale3}]  \par}
 
-These initial normated networks are not very good. This migth be what we expected, since we remove trivial information, but we still are able to improveon them quite a bit. Namely by using the exact model setups and training parameters from \ref{sec:setup} with one additional normalization layer before the first comparison value\footnote{You migth be quite a bit confused, why we chose other models but those that work well for unnormated networks to test our normalization, but this is just a problem of way to many just sligthly varying network setups: We used more quite different unnormated networks, but since learning zeros does not depend to much on network parameters, we simply use the new normated network setups for both networks, to not need to explain both.}.
+These initial normalized networks are not very good. This migth be what we expected, since we remove trivial information, but we still are able to improveon them quite a bit. Namely by using the exact model setups and training parameters from \ref{sec:setup} with one additional normalization layer before the first comparison value\footnote{You migth be quite a bit confused, why we chose other models but those that work well for unnormalized networks to test our normalization, but this is just a problem of way to many just sligthly varying network setups: We used more quite different unnormalized networks, but since learning zeros does not depend to much on network parameters, we simply use the new normalized network setups for both networks, to not need to explain both.}.
 Using this we are able to improve the network trained on top up to $0.377$.
 
-\subsubsection{Scaling in normated networks}\label{sec:scalenorm}
+\subsubsection{Scaling in normalized networks}\label{sec:scalenorm}
 
-Sadly this normalization does not chance scaling problems to much. Bigger networks still contain more trivial information, since the number of parameters fixed is constant, and even when using batches to scale, the invertibility is just a feature of the first batch
+Sadly this normalization does not change scaling problems to much. Bigger networks still contain more trivial information, since the number of parameters fixed is constant, and even when using batches to scale, the invertibility is just a feature of the first batch
 
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/m4scaleroc}
-\caption{(multi4scale roc) AUC values for higher normated batches by their training data}
+\caption{(multi4scale roc) AUC values for higher normalized batches by their training data}
 \label{fig:m4scaleroc}
   \end{figure}
 
@@ -1201,7 +1203,7 @@ Sadly this normalization does not chance scaling problems to much. Bigger networ
 
 \subsubsection{Improving the normalization even further}\label{sec:normplus}
 
-After seeing what an effect some kind of normalization can have, we are not completely satisfied anymore with the normated feature maps:
+After seeing what an effect some kind of normalization can have, we are not completely satisfied anymore with the normalized feature maps:
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.8\textwidth]{../imgs/aucmap928}
@@ -1219,7 +1221,7 @@ consider the highest $p_{T}$ Value (the lower rigth corner). While beeing the ge
   \end{figure}
 
 
-These values are basically constant, so its input it the same as the flag values (first collumn), from which we dont expect any physically useful information.
+These values are basically constant, so its input it the same as the flag values (first collumn), from which we don`t expect any physically useful information.
 So lets solve this: Since $lp_{T}$ mostly has the same structure\footnote{To be more precise, the difference between the first and the second particle is higher than the difference between the last two ones.}, most jets transverse momentum get divided by the first one, resulting in it always having the same value. We solve this by replacing the definition of $n$ in chapter \ref{sec:normalization} to be:
 \begin{equation}n = \frac{2 \cdot z}{\operatorname{max}{\left(\operatorname{abs}{\left(z \right)} \right)} + \operatorname{mean}{\left(\operatorname{abs}{\left(z \right)} \right)}}\end{equation}
 
@@ -1227,7 +1229,7 @@ removing the need to set one value to either positive or negative one, and thus
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.8\textwidth]{../imgs/aucmap534}
-\caption{ABE good norm}
+\caption{AUC feature map for a well normated network}
 \label{fig:aucmap534}
   \end{figure}
 
@@ -1241,16 +1243,16 @@ This we will explain in chapter \ref{sec:oneoff}.
 
 %from file ..\..\write\/data\06othernets\00intro
 \newpage
-\section{Solution 2: Mixed networks}\label{sec:secmixed}
+\section{Mixed networks}\label{sec:secmixed}
 {\scriptsize Referenced in: [\ref{sec:nobias}] [\ref{sec:evaloow}] [\ref{sec:secinv}] [\ref{sec:scaleloss}] [\ref{sec:invertibility}] [\ref{sec:usenorm}] [\ref{sec:decoding}] [\ref{sec:nogaegeneral}] [\ref{sec:oomath}]  \par}
 
 
 
 %from file ..\..\write\/data\06othernets\02oneoff
-\subsection{Oneoff networks }\label{sec:oneoff}
+\subsection{oneoff networks }\label{sec:oneoff}
 {\scriptsize Referenced in: [\ref{sec:imgmaps}] [\ref{sec:evaloow}] [\ref{sec:usenorm}] [\ref{sec:normplus}] [\ref{sec:mixedidea}] [\ref{sec:secfin}] [\ref{sec:quick2results}]  \par}
 
-When you consider the following feature map of a well normated network:
+When you consider the following feature map of a well normalized network:
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.8\textwidth]{../imgs/aucmapb}
@@ -1259,9 +1261,9 @@ When you consider the following feature map of a well normated network:
   \end{figure}
 
 
-You see, that most of the decision power is in the first feature, but the first feature, flag is basically just 1\footnote{Flag is 1 as long as the current event does not contain less particles than the network demands, and since this is a network with only 4 nodes, and there are very few jets with only 3 particles or even less, saying flag is a constant ($flag = 1$) is a quite good approximation.}. This migth seem a bit counterintuitive or unphysical at first, how can a variable without any physical meaning be a better seperator than those variables with physical meaning?
-To explain this, we need to take a bit more close look at what the network is doing: First, just because the output is has no physical meaning, this does not mean, that no physical variables are used in its calculcation. In fact, before this, we always just assumed that there is one parameter in the latent space, that is learned to be just a one from the input space\footnote{This is a bit of a simplification, most importantly it would be untestable, since instead of learning a constant, the network could learn a constant as a function of multiple parameters (for a simple example consider $x_{2} = x_{1} + 1$, both variables are not constant, making it harder to find this, but still $x_{2}$ has no additional information with respect to $x_{1}$, and there is a one learned as $- x_{1} + x_{2}$).}, but this distributiuon of decisionpower implies that this is not the case: If there would be a constant feature in the compression space, the constant output would be a trivial copy of this constant and thus have no physical meaning. More likely is the following: The network is able to reconstruct an $1$ from all the other parameters. This makes sence, since we got this AUC distribution by chancing the normalization in a way that made trivial ones in the input space much less likely\footnote{Since we stopped dividing by $\operatorname{max}{\left(\operatorname{abs}{\left(x \right)} \right)}$ and started dividing by $\left(\operatorname{max}{\left(\operatorname{abs}{\left(x \right)} \right)} + \operatorname{mean}{\left(\operatorname{abs}{\left(x \right)} \right)}\right) / 2$, it is no longer the case that there is either a $-1$ or a $1$ in each feature.}, and it also explains how an unphysical output can be physically useful: Since the are utilizing physical inputs, the resulting constant has to be a function of the inputs. And when you chance the inputs, the constant is also chanced and this chance we can use to differenciate signal and background events.
-And since this quality is better than every other autoencoder decision quality, it migth be useful to use this: If appearently nonphysical outputs can be at least as good as physical outputs, why not just use outputs that are nonphysical (Outputs that are one). This is what we call oneOff networks\footnote{Since the distance off 1 is the deciding quality indicator and it is a oneClass algorithm.}, and on paper it seams like a great idea: As shown before (see chapter \ref{sec:simplicity}), complexity is to a big part just width. You may be able to solve this by normalization, but this removes information, and oneOff networks would not require this\footnote{Since their output, $1$ , is obviously automatically normated.}\footnote{Also in practice it seems to be still a good idea to normate also oneoff networks, this migth be because this normalization also lets features the oneoff network focusses on to be more similar and thus easier to combine, or because similar sized inputs are easier to train on.}. Also there migth be a certain kind of complexity benefit, since the whole network is made to just minimize one distance\footnote{Actually, in practice it seems to simplify the training, if you dont use only one output, but multiple ones, that all are compared to 1 and which mean is used. This results in very high correlations in the outputs, but seems to help in the convergence of the network.} that is always the same, instead of optimizing some feature that migth be useful for some events, but weakenen it while considering other events, in which this feature plays a less important role. This should result in the network beeing able to learn more complicated functions.
+You see, that most of the decision power is in the first feature, but the first feature, flag is basically just one\footnote{Flag is 1 as long as the current event does not contain less particles than the network demands, and since this is a network with only 4 nodes, and there are very few jets with only 3 particles or even less, saying flag is a constant ($flag = 1$) is a quite good approximation.}. This migth seem a bit counterintuitive or unphysical at first, how can a variable without any physical meaning be a better seperator than those variables with physical meaning.
+To explain this, we need to take a bit more close look at what the network is doing: First, just because the output is has no physical meaning, this does not mean, that no physical variables are used in its calculcation. In fact, before this, we always just assumed that there is one parameter in the latent space, that is learned to be just a one from the input space\footnote{This is a bit of a simplification, most importantly it would be untestable, since instead of learning a constant, the network could learn a constant as a function of multiple parameters (for a simple example consider $x_{2} = x_{1} + 1$, both variables are not constant, making it harder to find this, but still $x_{2}$ has no additional information with respect to $x_{1}$, and there is a one learned as $- x_{1} + x_{2}$).}, but this distributiuon of decision power implies that this is not the case: If there would be a constant feature in the compression space, the constant output would be a trivial copy of this constant and thus have no physical meaning. More likely is the following: The network is able to reconstruct an $1$ from all the other parameters. This makes sence, since we got this AUC distribution by changing the normalization in a way that made trivial ones in the input space much less likely\footnote{Since we stopped dividing by $\operatorname{max}{\left(\operatorname{abs}{\left(x \right)} \right)}$ and started dividing by $\left(\operatorname{max}{\left(\operatorname{abs}{\left(x \right)} \right)} + \operatorname{mean}{\left(\operatorname{abs}{\left(x \right)} \right)}\right) / 2$, it is no longer the case that there is either a $-1$ or a $1$ in each feature.}, and it also explains how an unphysical output can be physically useful: Since the are utilizing physical inputs, the resulting constant has to be a function of the inputs. And when you change the inputs, the constant is also changed and this change we can use to differenciate signal and background events.
+And since this quality is better than every other autoencoder decision quality, it migth be useful to use this: If appearently nonphysical outputs can be at least as good as physical outputs, why not just use outputs that are nonphysical (Outputs that are one). This is what we call oneoff networks\footnote{Since the distance off 1 is the deciding quality indicator and it is a oneClass algorithm.}: As shown before (see chapter \ref{sec:simplicity}), complexity is to a big part just width. You may be able to solve this by normalization, but this removes information, and oneoff networks would not require this\footnote{Since their output, $1$ , is obviously automatically normalized.}\footnote{Also in practice it seems to be still a good idea to normate also oneoff networks, this migth be because this normalization also lets features the oneoff network focusses on to be more similar and thus easier to combine, or because similar sized inputs are easier to train on.}. Also there migth be a certain kind of complexity benefit, since the whole network is made to just minimize one distance\footnote{Actually, in practice it seems to simplify the training, if you don`t use only one output, but multiple ones, that all are compared to 1 and which mean is used. This results in very high correlations in the outputs, but seems to help in the convergence of the network.} that is always the same, instead of optimizing some feature that migth be useful for some events, but weakenen it while considering other events, in which this feature plays a less important role. This should result in the network beeing able to learn more complicated functions.
 
 We justify this idea mathematically in appendix \ref{sec:oomath} and \ref{sec:impro}
 
@@ -1269,13 +1271,13 @@ We justify this idea mathematically in appendix \ref{sec:oomath} and \ref{sec:im
 \subsubsection{oneoff quality}\label{sec:ooquality}
 
 
-So lets try this out: A simple dense network with just an output that should be one, sadly still has a lot of problems.
-First: the loss can go to basically zero($1 / 1000000000000$), which is a bit unphysical, since the loss, as a distance to one, is basically the variance of the used feature, and you would not expect there to be any physically significant feature of this accuracy in 4 particles\footnote{Especially, since the lowest difference there can be in the used float32 implementation is bigger than $1 / 100000000$ and thus, since the final loss is the mean of each loss, this would mean, that at least $0.9999$ of each event reproduce exactly 1.}. So there are features that are more trivial to learn, and make any decision process meaningless. And it is not neccesarily trivial to find those, there migth be those features that are just input variables of one (for example an input that would be set to flag), but not all of them are that easy to find. \footnote{A notable example migth be the preprocessing of $lp_{T}$. As descibed in chapter \ref{sec:data}, we used a preprocessing similar to that of particleNet: $x = \log{\left(p_{Tjet} / p_{T} \right)}$, but this means (because of the implementation), that a sum over $e^{- x}$ is always $1$. This migth be a good time to talk about functions in those kind of networks. Since we have to forbidden any biases (a bias would just result in the network learning a zero and adding a one as bias), the usual reason for a network to learn any function has to be modified a bit. Think about taylor approximations: A function like $e^{x}$ could be written as $1 + x + O\left(x^{2}\right)$ (with as many term as the networks needs), but for a network to learn $1$, the input of $e^{x}$ would then be learned to zero, the network would be one and it is basically the same as adding a constant bias. But adding a bias is not allowed, and thus the network can not learn $e^{x}$, but the network can learn $e^{x} - 1 = x + O\left(x^{2}\right)$, and, when $\operatorname{sum}{\left(e^{- x_{i}},i \right)} = 1$ then is $\operatorname{sum}{\left(-1 + e^{- x_{i}} \right)} = -3$ for 4 nodes, and thus the network can learn this, without having learned anything physically useful.}. This means, that training an oneoff network is a bit like outsmarting your algorithm. One thing that we found quite useful, is letting the network not only learn a one on the data that you are interrested in, but also zero on other random data.\footnote{We choose here random events with the same mean and standart deviation in each feature, as the original data, that still goes through the same preprocessing.}. When we use relu activations here\footnote{Activations are another thing where those networks can become trivial, think of a sigmoid and a network just learning infinite values before activation.}, learning values to be zero, means learning them just to be negative, and is thus way easier. This can demand that the network does not fixate on trivial features in the networksetup and preprocessing\footnote{Later on, in chapter \ref{sec:mixedidea}, this is no longer needed, and just complicates the training.}.
-A simple oneoff network reaches usually an auc of at best $0.6$ for the task of finding top jets, which is not to impressive. But if you look at the classification power as a function of the training epoch, you see that this only is so bad, since those AUC scores is way better at earlier epochs
+A simple dense network with just an output that should be one, sadly still has a lot of problems.
+First: the loss can go to basically zero($1 / 1000000000000$), which is a bit unphysical, since the loss, as a distance to one, is basically the variance of the used feature, and you would not expect there to be any physically significant feature of this accuracy in 4 particles\footnote{Especially, since the lowest difference there can be in the used float32 implementation is bigger than $1 / 100000000$ and thus, since the final loss is the mean of each loss, this would mean, that at least $0.9999$ of each event reproduce exactly 1.}. So there are features that are more trivial to learn, and make any decision process meaningless. And it is not neccesarily trivial to find those, there migth be those features that are just input variables of one (for example an input that would be set to flag), but not all of them are that easy to find. \footnote{A notable example migth be the preprocessing of $lp_{T}$. As descibed in chapter \ref{sec:data}, we used a preprocessing similar to that of particleNet: $x = \log{\left(p_{Tjet} / p_{T} \right)}$, but this means (because of the implementation), that a sum over $e^{- x}$ is always $1$. This migth be a good time to talk about functions in those kind of networks. Since we have to forbidden any biases (a bias would just result in the network learning a zero and adding a one as bias), the usual reason for a network to learn any function has to be modified a bit. Think about taylor approximations: A function like $e^{x}$ could be written as $1 + x + O\left(x^{2}\right)$ (with as many term as the networks needs), but for a network to learn $1$, the input of $e^{x}$ would then be learned to zero, the network would be one and it is basically the same as adding a constant bias. But adding a bias is not allowed, and thus the network can not learn $e^{x}$, but the network can learn $e^{x} - 1 = x + O\left(x^{2}\right)$, and, when $\operatorname{sum}{\left(e^{- x_{i}},i \right)} = 1$ then is $\operatorname{sum}{\left(-1 + e^{- x_{i}} \right)} = -3$ for 4 nodes, and thus the network can learn this, without having learned anything physically useful.}. This means, that training an oneoff network is a bit like outsmarting your algorithm. One thing that we found quite useful, is letting the network not only learn a one on the data that you are interrested in, but also zero on other random data.\footnote{We choose here random events with the same mean and standart deviation in each feature, as the original data, that still goes through the same preprocessing.}. When we use relu\footnote{A relu activation can be defined as $x + \operatorname{abs}{\left(x \right)}$. See Appendix \ref{sec:arelu} for why this is useful.}, learning values to be zero, means learning them just to be negative, and is thus way easier. This can demand that the network does not fixate on trivial features in the networksetup and preprocessing\footnote{Later on, in chapter \ref{sec:mixedidea}, this is no longer needed, and just complicates the training.}.
+A simple oneoff network reaches usually an auc of at best $0.6$ for the task of finding top jets, which is not to impressive. But if you look at the classification power as a function of the training epoch, you see that this only is so bad, since those AUC scores is way better at earlier epochs.
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.8\textwidth]{../imgs/mabe3}
-\caption{(multiabe 3)Auc as function of the epoch, trained on QCD, once for a graph oneOff and once for a dense oneOff. As you see, both relations show a maximum before the training ends, but the graph network is way more continuos}
+\caption{(multiabe 3)Auc as function of the epoch, trained on QCD, once for a graph oneoff and once for a dense oneoff. As you see, both relations show a maximum before the training ends, but the graph network is way more continuous}
 \label{fig:mabe3}
   \end{figure}
 
@@ -1284,8 +1286,8 @@ Sadly, this observation is not really useful, since stopping the training at the
 Another problem is again invertibility: It is possible to create an invertible oneoff network, but it is not trivially given. This becomes easier, when you use a lot of parameters. To do this, a graph network is less useful, than just a simple dense network.
 
 
-Even though they are not yet appliable here, we show in appendix \ref{sec:oomnist} that oneOff networks are very useful for finding anomalies in other datasets. This allows us to suggests that combining multiple oneOff retrains can increase the classification power even further.
-We also show that you can use oneOff networks to extract human readable information from physical events in appendix \ref{sec:oometrik}.
+Even though they are not yet appliable here, we show in appendix \ref{sec:oomnist} that oneoff networks are very useful for finding anomalies in other datasets. This allows us to suggests that combining multiple oneoff retrains can increase the classification power even further.
+We also show that you can use oneoff networks to extract human readable information from physical events in appendix \ref{sec:oometrik}.
 
 
 
@@ -1299,11 +1301,11 @@ We also show that you can use oneOff networks to extract human readable informat
 
 
 %from file ..\..\write\/data\06othernets\03idea
-\subsection{Compressed oneOff learning }\label{sec:mixedidea}
+\subsection{Compressed oneoff learning }\label{sec:mixedidea}
 {\scriptsize Referenced in: [\ref{sec:ooquality}]  \par}
 
-The main problem of autoencoder migth be the fact that its loss function is not neccesarily the best possible seperator\footnote{See chapter \ref{sec:losses} and chapter \ref{sec:secinv}.}, while the problem of oneOff networks seem to be that they focus on useless information, which keeps them from reaching their optimum\footnote{See chapter \ref{sec:oneoff}.}, but maybe combining both methods could solve both problems: You train an autoencoder to convert the input space into the latent space, to run an oneOff algorithm on this compressed space\footnote{We also tried alternative algorithms, but oneOff networks result in the best results, see for this appendix \ref{sec:mixedalt} and \ref{sec:other}.}. This means that the seperatorion function is now quite good, and the autoencoder can filter out trivial inputs. The idea of combining networks is not exactly new\footnote{See for example \cite{latentspace}.}. It also is harder to train, since we now have two independent networks: Something that improves the first network migth hurt the second, but in practice this works quite well.
-It is not yet clear if you want to train your autoencoder on the background data or on both the signal and the background. Here we train on background data, since every bit of higher inaccuracy that migth be reached by giving the original autoencoder unknown data, will help the following algorithm, but the effect of chancing this is tiny anyway. Also working as unsupervised as possible is not so easy: Defining a set with absolutely no anomalies is not completely unsupervised, but defining a set that is exactly half abnormal migth be worse: The anomalies we search are probably quite rare, and approximating this fraction as $0$ seems to be more realistic than approximating it as $0.5$.
+The main problem of autoencoder migth be the fact that its loss function is not neccesarily the best possible seperator\footnote{See chapter \ref{sec:losses} and chapter \ref{sec:secinv}.}, while the problem of oneoff networks seem to be that they focus on useless information, which keeps them from reaching their optimum\footnote{See chapter \ref{sec:oneoff}.}, but maybe combining both methods could solve both problems: You train an autoencoder to convert the input space into the latent space, to run an oneoff algorithm on this compressed space\footnote{We also tried alternative algorithms, but oneoff networks result in the best results, see for this appendix \ref{sec:mixedalt} and \ref{sec:other}.}. This means that the seperatorion function is now quite good, and the autoencoder can filter out trivial inputs. The idea of combining networks is not exactly new\footnote{See for example \cite{latentspace}.}. It also is harder to train, since we now have two independent networks: Something that improves the first network migth hurt the second, but in practice this works quite well.
+It is not yet clear if you want to train your autoencoder on the background data or on both the signal and the background. Here we train on background data, since every bit of higher inaccuracy that migth be reached by giving the original autoencoder unknown data, will help the following algorithm, but the effect of changing this is tiny anyway. Also working as unsupervised as possible is not so easy: Defining a set with absolutely no anomalies is not completely unsupervised, but defining a set that is exactly half abnormal migth be worse: The anomalies we search are probably quite rare, and approximating this fraction as $0$ seems to be more realistic than approximating it as $0.5$.
 
 
 
@@ -1316,7 +1318,7 @@ It is not yet clear if you want to train your autoencoder on the background data
 {\scriptsize Referenced in: [\ref{sec:decoding}]  \par}
 
 With the same setup as before (see chapter \ref{sec:setup}) and normation as well as after training 25 oneoff networks on each latent space we gain the final top tagger for this thesis
-\subsubsection{trained on QCD}\label{sec:classQCD}
+\subsubsection{Trained on QCD}\label{sec:classQCD}
 
 
 \begin{figure}[H]
@@ -1330,7 +1332,7 @@ With the same setup as before (see chapter \ref{sec:setup}) and normation as wel
   \includegraphics[width=.8\linewidth]{../imgs/seproc928}
   \label{fig:sephist928_2}
 \end{subfigure}%
-\caption{oneoff loss distribution and roc curve for a network trained on top jets}
+\caption{Oneoff loss distribution and roc curve for a network trained on top jets}
 \label{fig:sephist928}
 \end{figure}
 
@@ -1352,7 +1354,7 @@ Interrestingly this also helps the reconstruction quality
   \includegraphics[width=.8\linewidth]{../imgs/ptdraw928}
   \label{fig:simpledraw928_2}
 \end{subfigure}%
-\caption{Reconstruction images for a normated network trained on QCD}
+\caption{Reconstruction images for a normalized network trained on QCD}
 \label{fig:simpledraw928}
 \end{figure}
 
@@ -1361,7 +1363,7 @@ Interrestingly this also helps the reconstruction quality
 
 
 
-\subsubsection{trained on top}\label{sec:classtop}
+\subsubsection{Trained on top}\label{sec:classtop}
 
 
 Trained on top this improves quite a lot.
@@ -1397,7 +1399,7 @@ Also here the reconstruction quality improves
   \includegraphics[width=.8\linewidth]{../imgs/ptdraw1128}
   \label{fig:simpledraw1128_2}
 \end{subfigure}%
-\caption{Reconstruction images for a normated network trained on top}
+\caption{Reconstruction images for a normalized network trained on top}
 \label{fig:simpledraw1128}
 \end{figure}
 
@@ -1407,10 +1409,10 @@ Also here the reconstruction quality improves
 
 
 %from file ..\..\write\/data\06othernets\06scale
-\subsection{Scale }\label{sec:scale3}
+\subsection{Scaling for oneoff networks }\label{sec:scale3}
 {\scriptsize Referenced in: [\ref{sec:ldm}]  \par}
 
-OneOffs still dont solve the problem of different parts of the network beeing added supoptimally. You see this when you consider this 9 node network trained on top jets:
+oneoffs still don`t solve the problem of different parts of the network beeing added supoptimally. You see this when you consider this 9 node network trained on top jets:
 \begin{figure}[H]
 \begin{subfigure}{0.45\textwidth}
   \centering
@@ -1439,7 +1441,8 @@ On the other hand, the batches considered in chapter \ref{sec:normimpro} are now
   \end{figure}
 
 
-Here you see a much more interresting relation compared to before. The variance grows with the batch index, which is expected, but some networks actually beat the AUC score of the first batch (batch 3 has a event below $0.15$). This is a result of the number of particles in each jet becoming a feature at some point. You see this, by noticing that the relation between auc and batch number is not linear: The AUCs for the second batch migth even be some of the worst, even though they should have the second most information next the the first batch.
+Here you see a much more interresting relation compared to before. The variance grows with the batch index, which is expected, but some networks actually beat the AUC score of the first batch (batch 3 has a event below $0.15$). This is a result of the number of particles in each jet becoming a feature at some point. You see this, by noticing that the relation between AUC and batch number is not linear: The AUCs for the second batch migth even be some of the worst, even though they should have the second most information next the the first batch.
+
 
 %from file ..\..\write\/data\06othernets\08compare
 
@@ -1449,15 +1452,15 @@ Here you see a much more interresting relation compared to before. The variance
 
 %from file ..\..\write\/data\07otherdata\01intro
 \newpage
-\section{Other data}\label{sec:secdata}
+\section{Applying this model to other datasets}\label{sec:secdata}
 {\scriptsize Referenced in: [\ref{sec:data}] [\ref{sec:evaloow}] [\ref{sec:simplicity}] [\ref{sec:invertibility}] [\ref{sec:secfin}] [\ref{sec:future}] [\ref{sec:nogaegeneral}]  \par}
 
 We migth be able to supervisedly seperate probably any kind of data, but as shown in the previous chapters, if we remove the labels, this exercise becomes a lot harder. Still, usually this problem is stated as follows: Given a set of datapoints, can we write an algorithm to detect a second set of datapoints. The only difference to the supervised case is the fact that we cannot look at the anomaly set in training. And information from the validation dataset can leak into the model setup\footnote{See \cite{leakage}.}.
-This is an effect, that is usually solved by introducing test data. Data that is only used once at the end of your analysis: If your network works worse on this data, then your setup contains information about your validation data and is no longer as general. We want to use this chapter to introduce our test data. The difference here is, that we cannot simply use some part of our validation set: It are not the events of the validation set that are leaking\footnote{As shown in \ref{sec:evalae} our networks dont overfit.}, but the specifics of your anomalies. So as test set, we have to use completely different anomalies\footnote{Training on your anomalies to find your background can help, but even this can not really exclude that your data leaks into your model( if your signal and background differ completely in one parameter, optimizing both ways would result in a network only focussing on this parameter, and thus not beeing very general), so the only real way migth be to train on as much datasets as possible, and demand that all work.}.
+This is an effect, that is usually solved by introducing test data. Data that is only used once at the end of your analysis: If your network works worse on this data, then your setup contains information about your validation data and is no longer as general. We want to use this chapter to introduce our test data. The difference here is, that we cannot simply use some part of our validation set: It are not the events of the validation set that are leaking\footnote{As shown in \ref{sec:evalae} our networks don`t overfit.}, but the specifics of your anomalies. So as test set, we have to use completely different anomalies\footnote{Training on your anomalies to find your background can help, but even this can not really exclude that your data leaks into your model( if your signal and background differ completely in one parameter, optimizing both ways would result in a network only focussing on this parameter, and thus not beeing very general), so the only real way migth be to train on as much datasets as possible, and demand that all work.}.
 
 
-You could see this, as chancing our initial task: Instead of finding one specific anomaly, we now want to find every other anomaly.
-Please take a moment to notice the huge difference in complexity of this task: Defining every alternative dataset as signals is not solved by looking at any attribute to differentiate datasets: There will always be another dataset, that is entirely the same as the background set if you are looking at this attribute only. Also there will even be a dataset, that has an attribute that looks more than the background than the actual background\footnote{Looks the same as the background but with lower width.}. You could say, that finding all alternative datasets, is more about defining your background, than about finding differences. But as even oneOff networks, that are designed to define your background datasets, in theory are expected to be able to be trapped in some features\footnote{See appendix \ref{sec:impro}.}, the only way to truly evaluate an algorithm, is experimentally. And since we cannot generate all alternative datasets and we have to work with comparing two datasets to each other. But we can at least give some sence of generality to the networks, by looking at different kinds of datasets.
+You could see this, as changing our initial task: Instead of finding one specific anomaly, we now want to find every other anomaly.
+One should notice the huge difference in complexity of this task: Defining every alternative dataset as signals is not solved by looking at any attribute to differentiate datasets: There will always be another dataset, that is entirely the same as the background set, if you are looking at this attribute only. Also there will even be a dataset, that has an attribute that looks more than the background than the actual background\footnote{Looks the same as the background but with lower width.}. You could say, that finding all alternative datasets, is more about defining your background, than about finding differences. But as even oneoff networks, that are designed to define your background datasets, in theory are expected to be able to be trapped in some features\footnote{See appendix \ref{sec:impro}.}, the only way to truly evaluate an algorithm, is experimentally. And since we cannot generate all alternative datasets and we have to work with comparing two datasets to each other. But we can at least give some sence of generality to the networks, by looking at different kinds of datasets.
 
 
 
@@ -1466,7 +1469,7 @@ Please take a moment to notice the huge difference in complexity of this task: D
 \subsection{Ligth dark matter }\label{sec:ldm}
 {\scriptsize Referenced in: [\ref{sec:leptons}] [\ref{sec:crossdata}] [\ref{sec:oofail}]  \par}
 
-This set of datapoints is generated by thorben finke and consists out of jets of transverse momentum between $150 \cdot GeV$ and $270 \cdot Gev$ of either QCD jets, or those initiated by a dark matter candidate sugested in \cite{ldm}.
+This set of datapoints is generated by thorben finke and consists out of jets of transverse momentum between $150 \cdot GeV$ and $270 \cdot GeV$ of either QCD jets, or those initiated by a dark matter candidate sugested in \cite{ldm} (ldm data).
 This dataset implies a unsupervised classification task that is way more difficult than the usual top tagging, and as we will see, even more complicated than the other datasets that we test our algorithm on here. 
 The first thing that makes this dataset so much more complicated is the angular distribution: while you can use this distribution to differentiate top jets from their QCD counterparts alone, and this quite well (see chapter \ref{sec:simplicity}), here both angular distributions are basically the same
 \begin{figure}[H]
@@ -1507,29 +1510,29 @@ That beeing said, there is one easily understandable parameter that can be used
 
 
 
-Sadly, this parameter is not very useful because of two reasons
+Sadly, this parameter is not very useful because of two reasons:
 \begin{itemize}
 
-    \item Our graph based networks, especcially the ones we talk about in this chapter only have 4 particles. That means that the number of particles just does not enter the network at all\footnote{To be precise, there are o(10) jets with less than 4 particles in our dataset, so it actually is inputted, but only to a neglicible amount.}, and even though we can increase the number of particles that enter the network, this will result in less well trained networks (see chapter \ref{sec:scale3}), and at the end</i>
-    \item 
-We actually dont want to have a network that just focusses on the number of nodes, as this is a fairly weak way of differentiating jets, resulting only in $O\left(1\right)$ Auc values in the best case. And even though this is way better than any classification score that we achieve in the following, focussing on only one parameter looses every sence of generality
+    \item Our graph based networks, especcially the ones we talk about in this chapter only have 4 particles. That means that the number of particles just does not enter the network at all\footnote{To be precise, there are o(10) jets with less than 4 particles in our dataset, so it actually is inputted, but only to a neglicible amount.}, and even though we can increase the number of particles that enter the network, this will result in less well trained networks (see chapter \ref{sec:scale3}), and at the end
+
+    \item We actually don`t want to have a network that just focusses on the number of nodes, as this is a fairly weak way of differentiating jets, resulting only in $O\left(1\right)$ Auc values in the best case. And even though this is way better than any classification score that we achieve in the following, focussing on only one parameter looses every sence of generality
 
 
 \end{itemize}
 
-This means, that we train 4 node networks\footnote{The low number of particles becomes a benefit, since we can be sure not to use the particle number.}, that hopefully find some sense of substructre, that makes it possible to differentiate those jets. Sadly this means also, that every result has to be fairly bad! Why? The problem is generality: There is a dataset with different substructure, but there is also another dataset with completely different angular distribution, and since we want our networks to find both, this also means, that having the same angular distribution has some effect on the network making it more likely that this is actually the same datatyp. There migth be still some different substructure, but what would you give more importance to?There are three effects here
+This means, that we train 4 node networks\footnote{The low number of particles becomes a benefit, since we can be sure not to use the particle number.}, that hopefully find some sense of substructre, that makes it possible to differentiate those jets. Sadly this means also, that every result has to be fairly bad: There is a dataset with different substructure, but there is also another dataset with completely different angular distribution, and since we want our networks to find both, this also means, that having the same angular distribution has some effect on the network making it more likely that this is actually the same datatyp. There migth be still some different substructure, butthere are three effects here:
 \begin{itemize}
 
-    \item your first thougth migth be, that this relative uncertainity is not affected by the function complexity, and it is true, that if you got only two variables, their effect is completely unaffected by it, but there is a catch
+    \item Your first thougth migth be, that this relative uncertainity is not affected by the function complexity, and it is true, that if you got only two variables, their effect is completely unaffected by it, but there is a catch.
 
-    \item first, you cannot assume each variable to be exactly known, and a sligth variation in a complex formula can have a much bigger effect than in an easy formula (thing of a momentum 4 vector representing an electron: the formula getting the energy from this 4 vector is much more stable under variations of the 4 momentum, than the formula getting its mass)
+    \item You cannot assume each variable to be exactly known, and a sligth variation in a complex formula can have a much bigger effect than in an easy formula (thing of a momentum 4 vector representing an electron: the formula getting the energy from this 4 vector is much more stable under variations of the 4 momentum, than the formula getting its mass).
 
-    \item Secondly, the number of neurons is finite, and you could argue, that the number of calculatable variables is not. This means that the network has to choose favorites. And since every given feature can be weigthedly added in a way that (at least for a tiny amount) improves the current relative uncertainity, choosing a complicated/expensive feature also means, not choosing more less complicated features
+    \item Also the number of neurons is finite, and you could argue, that the number of calculatable variables is not. This means that the network has to choose favorites. And since every given feature can be weigthedly added in a way that (at least for a tiny amount) improves the current relative uncertainity, choosing a complicated/expensive feature also means, not choosing more less complicated features.
 
 
 \end{itemize}
 
-This means, that there is a sligth preference of oneOff networks, to choose easier features\footnote{In general, this is actually a really good thing: Not only does is this statistically useful, but this also means, that oneOffs have a build in regulator, that prevents them from overfitting (at least to a degree), and thus they are a bit more general.}, which means, that this is a really hard test for them, and the only thing that we can realistically demand here is invertibility: A Network, trained on ligth QCD jets, that thinks ldm jets are more complicated as well as the inverse.
+This means, that there is a sligth preference of oneoff networks, to choose easier features\footnote{In general, this is actually a really good thing: Not only does is this statistically useful, but this also means, that oneoffs have a build in regulator, that prevents them from overfitting (at least to a degree), and thus they are a bit more general.}, which means, that this is a really hard test for them, and the only thing that we can realistically demand here is invertibility: A Network, trained on ligth QCD jets, that thinks ldm jets are more complicated as well as the inverse.
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/multisep10}
@@ -1538,7 +1541,7 @@ This means, that there is a sligth preference of oneOff networks, to choose easi
   \end{figure}
 
 
-As you see, this is not at all trivial, but when we consider the loss of the oneOff network, which is drawn on the x axis as quality, the best networks are actually invertible. And since this is still completely unsupervised, just using the feature quality of a network, we can say that we can generate invertible anomaly detection algorithms on this dataset. 
+As you see, this is not at all trivial, but when we consider the loss of the oneoff network, which is drawn on the x axis as quality, the best networks are actually invertible. And since this is still completely unsupervised, just using the feature quality of a network, we can say that we can generate invertible anomaly detection algorithms on this dataset. 
 That beeing said, this is obviously not useful at all: half a percentage in AUC does not help you at differentiating new physics, but it is worth to note, that also top tagging after normalization looked very similar to this once (see chapter \ref{sec:invertibility}), and seeing that there the classification quality improved a lot, we see no reason, why you could not improve and optimize this network to be drastically better. Especcially, since we did not run any hyperparameter optimization (except for the compression size, which is 1 bigger (at 10)\footnote{We think, that this higher compression size allows the network to understand more subfeatures.}), and still only use 4 particles
 
 
@@ -1551,10 +1554,10 @@ That beeing said, this is obviously not useful at all: half a percentage in AUC
 
 
 
-\subsubsection{Quark v gluon}\label{sec:qg}
+\subsubsection{Quark or gluon}\label{sec:qg}
 
 
-Quark and gluon data, is generated by madgraph\cite{madgraph}, Pythia\cite{pythia} and delphes\cite{delphes}. One set is generated as parton parton to gluon gluon collisions and another as parton parton to two parton without gluon collisions. Jets are used, if there transverse jet momentum is between 550 and 650 geV. This data was used originally to see if a QCD trained classifier makes a easily accessible difference between quarks and gluons\footnote{You could interpret this, as another form of complexity: while top jets are all the result of top quarks, with QCD jets there are multiple options, we though this could explain why QCD trained encoder are generally worse, but this just not the case.}, but even though this is seems not to be the case, we can still use this dataset to test our algorithm a bit further. Again we use 4 particle networks, with a compression size of $9$ and only neglicible hyperparameter optimization to reach quality of
+Quark and gluon data, is generated by madgraph\cite{madgraph}, Pythia\cite{pythia} and delphes\cite{delphes}. One set is generated as parton parton to gluon gluon collisions and another as parton parton to two parton without gluon collisions. Jets are used, if there transverse jet momentum is between 550 and 650 geV. This data was used originally to see if a QCD trained classifier makes a easily accessible difference between quarks and gluons\footnote{You could interpret this, as another form of complexity: while top jets are all the result of top quarks, with QCD jets there are multiple options, we though this could explain why QCD trained encoder are generally worse, but this just not the case.}, but even though this is seems not to be the case, we can still use this dataset to test our algorithm a bit further. Again we use 4 particle networks, with a compression size of $9$ and only neglicible hyperparameter optimization to reach quality of sligthly above random.
 \begin{figure}[H]
 \begin{subfigure}{0.45\textwidth}
   \centering
@@ -1575,12 +1578,12 @@ Quark and gluon data, is generated by madgraph\cite{madgraph}, Pythia\cite{pythi
 
 As you see, these are invertible networks, and even though they are not very good ones, as described in the previous chapter \ref{sec:ldm} this does not really matter, since optimization has the potential to improve them quite a lot. \cite{quarkdata} could be seen as a reference paper for this process, even though they use a supervised approach and high level input data on different transverse momentum ranges, their achieves AUC values below $0.9$ suggest that this tagging job is more complicated than the usual top tagging. Also chapter \ref{sec:crossdata} will support this hypothesis. 
 
-\subsubsection{leptons}\label{sec:leptons}
+\subsubsection{Leptons}\label{sec:leptons}
 
 
 
 
-This dataset is not very physically useful, and more interresting from an anomaly detection standpoint: We again generate particle collisions using madgraph, Pythia and delphes, but instead of partons colliding into partons, we use leptons colliding and producing partons. For the first set, we use any combination of electrons and muons with arbitrary charge, and for the second one we only use tau leptons. We also use a fairly big transverse momentum range for the jet of $20 \cdot Gev$ to $5000 \cdot Gev$ to see if our algorithm is affected by this bigger range.
+This dataset is not very physically useful, and more interresting from an anomaly detection standpoint: We again generate particle collisions using madgraph, Pythia and delphes, but instead of partons colliding into partons, we use leptons colliding and producing partons. For the first set, we use any combination of electrons and muons with arbitrary charge, and for the second one we only use tau leptons. We also use a fairly big transverse momentum range for the jet of $20 \cdot GeV$ to $5000 \cdot GeV$ to see if our algorithm is affected by this bigger range.
 
 \begin{figure}[H]
 \begin{subfigure}{0.45\textwidth}
@@ -1619,10 +1622,10 @@ Again you see a clear invertibility, helping to support the suggested generality
 
 
 %from file ..\..\write\/data\07otherdata\05cross
-\subsection{Even more comparison }\label{sec:crossdata}
+\subsection{Cross comparisons }\label{sec:crossdata}
 {\scriptsize Referenced in: [\ref{sec:leptons}]  \par}
 
-At the beginning of this chapter, we called anomaly detection the task of finding everything that is not similar to the trained on class. And even though we tried to evaluate this task, by showing the invertibility on a multitude of datasets, we slowly are out of particles to test it on\footnote{Especcially since the initial toptagging dataset already contained the whole of QCD.}. That beeing said, one thing we did not yet do, is to mix the datasets. You migth question how useful this is from a physical standpoint, as there will probably never be a situation, in which you want to find leptons, only knowing gluons. The point is that new physics could have a neirly arbitrary form, and even though we will never live in world in which we only know about gluons, finding data that does not look like gluons is very useful. We think that these experiments introduce thrust in the algorithm used, as chapter \ref{sec:secinv} clearly shows, that invertibility and feature triviality can be linked.  And since training unsupervised does even mean, that we dont have to train new anomaly detection models, there is no reason not to compare those jets
+At the beginning of this chapter, we called anomaly detection the task of finding everything that is not similar to the trained on class. And even though we tried to evaluate this task, by showing the invertibility on a multitude of datasets, we slowly are out of particles to test it on\footnote{Especcially since the initial toptagging dataset already contained the whole of QCD.}. That beeing said, one thing we did not yet do, is to mix the datasets. You migth question how useful this is from a physical standpoint, as there will probably never be a situation, in which you want to find leptons, only knowing gluons. The point is that new physics could have a neirly arbitrary form, and even though we will never live in world in which we only know about gluons, finding data that does not look like gluons is very useful. We think that these experiments introduce thrust in the algorithm used, as chapter \ref{sec:secinv} clearly shows, that invertibility and feature triviality can be linked.  And since training unsupervised does even mean, that we don`t have to train new anomaly detection models, there is no reason not to compare those jets.
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/crosssep}
@@ -1632,7 +1635,7 @@ At the beginning of this chapter, we called anomaly detection the task of findin
 
 
 
-As you see, there are only few spots that are not invertible (we chanced the meaning of each AUC value, in such a way, that each slot should be deeply blue in the best case). For simplicity we mark the noninvertible networks with black spots, but this still does not allow to see every value, so here are those comparisons again as a table
+As you see, there are only few spots that are not invertible (we changed the meaning of each AUC value, in such a way, that each slot should be deeply blue in the best case). For simplicity we mark the noninvertible networks with black spots, but this still does not allow to see every value, so here are those comparisons again as a table.
 
 
 
@@ -1669,9 +1672,9 @@ lept & $0.6254$ & $0.7018$ & $0.6753$ & $0.6595$ & $0.5647$ & $0.6283$ & $0.6023
 \label{table:crossinv}
 \end{table}
 
-As you see, everything is either invertible, or at least very close. Furthermore, only 4 noninvertible comparisons exist, and are always less than $2.5$\% worse than random guessing, and are trained on ligth dark matter or ligth QCD data, which as explained in \ref{sec:ldm} is hard to differentiate\footnote{It is a bit weird, that ligth QCD jets are more similar to ligth dark matter data, than to QCD of higher energy, especially since every value is normated quite thorougly (chapter \ref{sec:normalization}), but this still does not mean, that their is something wrong with the data, as the normalization has no effect on they number of particles, but just on the size of each value. And since higher energy jets can decay differently, this migth explain why ligth QCD and ligth dark matter jets look so similar.}.
+As you see, everything is either invertible, or at least very close. Furthermore, only 4 noninvertible comparisons exist, and are always less than $2.5$\% worse than random guessing, and are trained on ligth dark matter or ligth QCD data, which as explained in \ref{sec:ldm} is hard to differentiate\footnote{It is a bit weird, that ligth QCD jets are more similar to ligth dark matter data, than to QCD of higher energy, especially since every value is normalized quite thorougly (chapter \ref{sec:normalization}), but this still does not mean, that their is something wrong with the data, as the normalization has no effect on they number of particles, but just on the size of each value. And since higher energy jets can decay differently, this migth explain why ligth QCD and ligth dark matter jets look so similar.}.
 
-Also please note, that the rows and collumns that are related to top jets are clearly visible: It seems to be a much easier task to differentiate top jets, than every other dataset, even when we use normated networks. This suggests that only using top jets as anomalies artificially inflates your performance.
+Also note, that the rows and collumns that are related to top jets are clearly visible: It seems to be a much easier task to differentiate top jets, than every other dataset, even when we use normalized networks. This suggests that only using top jets as anomalies artificially inflates your performance.
 
 
 
@@ -1711,7 +1714,7 @@ THNKING ABOUT SOMETHING LIKE ABOUT WRITING STUFF JUST DOES NOT WORK, THIS LIST W
 
     \item Actual image like losses (calculate histograms and compare them)
 
-        \item permutation invariant losses, we already tried some, but they dont yet work well, even though they theoretically should
+        \item permutation invariant losses, we already tried some, but they don`t yet work well, even though they theoretically should
 
         \item L1.5
 
@@ -1733,7 +1736,7 @@ THNKING ABOUT SOMETHING LIKE ABOUT WRITING STUFF JUST DOES NOT WORK, THIS LIST W
 
   \begin{itemize}
 
-    \item It seems to be easier to work on transformed 4 momenta, but on classical 4 vectors, explain and chance this
+    \item It seems to be easier to work on transformed 4 momenta, but on classical 4 vectors, explain and change this
 
         \item add lowly correlated variables like the mass
 
@@ -1781,11 +1784,11 @@ THNKING ABOUT SOMETHING LIKE ABOUT WRITING STUFF JUST DOES NOT WORK, THIS LIST W
 
     \item no more nans
 
-        \item improve networks that dont use a latent space of just one node
+        \item improve networks that don`t use a latent space of just one node
 
 
 \end{itemize}
-      \item Use other scores then AUCs, and try to optimize oneOff networks for them
+      \item Use other scores then AUCs, and try to optimize oneoff networks for them
 
       \item explain why better compression algorithm work worse
 
@@ -1819,7 +1822,7 @@ These implementations are usually defined by an encoding and a decoding algorith
 
 \subsubsection{trivial models}\label{sec:failedtrivial}
 
-Let us start with the probably most simple autoencoder algorithms: To make a $n$ node graph into a $m$ node graph, we just cut away the last nodes until there are only $m$ nodes left\footnote{Please note the importance of the $p_{T}$ ordering here: Cutting the last particles means cutting the particles with lowest $p_{T}$ and thus the probably least important particles.} to reduce the graph size, and add zero valued particles to it again. One difficulty here lies in the fact that those particles have no more graph connections, this we solved by just keeping the original graph connections stored. Sadly, those networks just dont work: even when we would set the compression size over the input size, the reproduced jets hardly bare any resemble to the input jets: This is the first example of the central problem of graph autoencoding: Permutation invariance. Consider the following encoder: two numbers $a$ and $b$ where $a = b + 1$, this would be trivial to compress into one number for a normal\footnote{Dense.} Autoencoder(maybe just take $a$), but here we have to respect permutaion symmetry, so basically we do not know what the first and what the second particle is and how do we decompress now? In this context you could keep one of the parameters and try to encode if the other one is bigger or smaller than this, maybe you also know that $0 \leq a$ and you could multiply it by $-1$ if it is the smaller one, but this is less than trivial, and by increasing the number of parameters this gets even more complicated. This is a problem that mostly appears as the inability of even a "good" Autoencoder to work with and compression size that is equal to the input size, building an identity (see chapter \ref{sec:identities}). 
+Let us start with the probably most simple autoencoder algorithms: To make a $n$ node graph into a $m$ node graph, we just cut away the last nodes until there are only $m$ nodes left\footnote{Please note the importance of the $p_{T}$ ordering here: Cutting the last particles means cutting the particles with lowest $p_{T}$ and thus the probably least important particles.} to reduce the graph size, and add zero valued particles to it again. One difficulty here lies in the fact that those particles have no more graph connections, this we solved by just keeping the original graph connections stored. Sadly, those networks just don`t work: even when we would set the compression size over the input size, the reproduced jets hardly bare any resemble to the input jets: This is the first example of the central problem of graph autoencoding: Permutation invariance. Consider the following encoder: two numbers $a$ and $b$ where $a = b + 1$, this would be trivial to compress into one number for a normal\footnote{Dense.} Autoencoder(maybe just take $a$), but here we have to respect permutaion symmetry, so basically we do not know what the first and what the second particle is and how do we decompress now? In this context you could keep one of the parameters and try to encode if the other one is bigger or smaller than this, maybe you also know that $0 \leq a$ and you could multiply it by $-1$ if it is the smaller one, but this is less than trivial, and by increasing the number of parameters this gets even more complicated. This is a problem that mostly appears as the inability of even a "good" Autoencoder to work with and compression size that is equal to the input size, building an identity (see chapter \ref{sec:identities}). 
 That beeing said, permutation invariance can also be a benefit, especially in permutation invariant input data, more to this in chapter \ref{sec:aperminv}
 
 
@@ -1866,7 +1869,7 @@ K neirest neighbour is the first algorithm that improves over simply using the a
 
 \subsubsection{oneoff}\label{sec:mixedoo}
 
-Oneoffs seem to be the way to go here, and will be used exclusively for the rest of this chapter: One network reached about $0.247$ with an error of $0.005$, already beating all all our autoencoder, and by combining multiple ones, you can reach an AUC of about $0.2$ beeing quite good.
+oneoffs seem to be the way to go here, and will be used exclusively for the rest of this chapter: One network reached about $0.247$ with an error of $0.005$, already beating all all our autoencoder, and by combining multiple ones, you can reach an AUC of about $0.2$ beeing quite good.
 
 
 \begin{figure}[H] 
@@ -1900,7 +1903,7 @@ As you see, this autoencoder takes QCD jets, transforms them as introduced in \r
 
 \subsubsection{topK}\label{sec:quicktopK}
 
-The probably most comonly used algorithm, to construct a set of graph connections from a list of vectors, topK, seems to be quite easy to understand: you connect each vector, to the $K$ vectors that are most similar to it. The difficulty lies in the word similar: Here two vectors are more similar, the smaller the l2 difference is. In an attempt, to make this more powerful, we also use a learnable metrik in this l2 difference. Even though this migth not be strictly neccesary, since the network can chance parameters to accomodate its sence of similarity, this still allows the network to better choose what to focus on in each topK layer. It can be quite useful for autoencoder, since for example ignoring a parameter, could else only be done, by decreasing its size in relation to the other parameters, which migth not be optimal, when you want an accurate reproduction. This also allows you to create a graph, before having any learnable layers. On the other hand, these metrik can complicate the calculcation of the adjacency matrix, which we try to manage by demanding that the metrik is entirely diagonal, reducing also the needed time drastically, and the parameters of the metrik can increase the occurence of divergences in training, since even a small chance of those parameters can effect the network output in huge ways. That beeing said, having a humanly understandable metrik, can lead to interresting insigths (see appendix \ref{sec:ametrikana}).
+The probably most comonly used algorithm, to construct a set of graph connections from a list of vectors, topK, seems to be quite easy to understand: you connect each vector, to the $K$ vectors that are most similar to it. The difficulty lies in the word similar: Here two vectors are more similar, the smaller the l2 difference is. In an attempt, to make this more powerful, we also use a learnable metrik in this l2 difference. Even though this migth not be strictly neccesary, since the network can change parameters to accomodate its sence of similarity, this still allows the network to better choose what to focus on in each topK layer. It can be quite useful for autoencoder, since for example ignoring a parameter, could else only be done, by decreasing its size in relation to the other parameters, which migth not be optimal, when you want an accurate reproduction. This also allows you to create a graph, before having any learnable layers. On the other hand, these metrik can complicate the calculcation of the adjacency matrix, which we try to manage by demanding that the metrik is entirely diagonal, reducing also the needed time drastically, and the parameters of the metrik can increase the occurence of divergences in training, since even a small change of those parameters can effect the network output in huge ways. That beeing said, having a humanly understandable metrik, can lead to interresting insigths (see appendix \ref{sec:ametrikana}).
 You could ask yourself, if a topK algorithm is the best choice, since the number of possible adjacency matrices is quite low, see for this appendix \ref{sec:atopkwhy}.
 Finally, it should be noted, that the topK layer can increase the size of each of the feature vectors, which is useful for the compression algorithm, even though in this specific example this is not used.
 
@@ -1935,7 +1938,7 @@ Another thing that has to be clearified concerning this model, is the training p
   \end{figure}
 
 
-As you see, there is not really any progress made in the training\footnote{Except for maybe the first epoch, which is not shown in these kind of plots.}, but you already see one fact, that will be quite common in the following: The validation loss is not (much) bigger than the training loss, neither at the end, not anywhere. This is fairly uncommon, as usually earlyStopping is used to compat overfitting, and validation losses that seem to increase at some point, but also easily explained, since encoder and decoder only amount to a total of 840 trainable parameters, which is not enough to store informations for $O\left(1\right)$ events. Interrestingly, this seems to be a clear benefit for graph autoencoder, as even bigger networks with similar amounts of parameters, trained on less data, dont seem to show any tendency to overfit. This allows us to reduce the training size to at least 2 orders of magnitude less, without any quality loss (see chapter \ref{sec:asize}), and you could even ask yourself if it would not be possible to remove the whole need of splitting your data into training and validation data. That beeing said, this dataseperation is mentained for the rest of the thesis, and this overfitting safety comes at a price: the validation loss migth not increase in relation to the training loss, but that does not mean that both cannot increase in parallel. This, and the fact that graph training curves are way more noisy than usual training curves, make earlyStopping still a viable training callback, and result in most of the reasons, each training stops.
+As you see, there is not really any progress made in the training\footnote{Except for maybe the first epoch, which is not shown in these kind of plots.}, but you already see one fact, that will be quite common in the following: The validation loss is not (much) bigger than the training loss, neither at the end, not anywhere. This is fairly uncommon, as usually earlyStopping is used to compat overfitting, and validation losses that seem to increase at some point, but also easily explained, since encoder and decoder only amount to a total of 840 trainable parameters, which is not enough to store informations for $O\left(1\right)$ events. Interrestingly, this seems to be a clear benefit for graph autoencoder, as even bigger networks with similar amounts of parameters, trained on less data, don`t seem to show any tendency to overfit. This allows us to reduce the training size to at least 2 orders of magnitude less, without any quality loss (see chapter \ref{sec:asize}), and you could even ask yourself if it would not be possible to remove the whole need of splitting your data into training and validation data. That beeing said, this dataseperation is mentained for the rest of the thesis, and this overfitting safety comes at a price: the validation loss migth not increase in relation to the training loss, but that does not mean that both cannot increase in parallel. This, and the fact that graph training curves are way more noisy than usual training curves, make earlyStopping still a viable training callback, and result in most of the reasons, each training stops.
 
 
 \subsubsection{Results}\label{sec:quickres1}
@@ -1981,7 +1984,7 @@ If we look at the featuremap, the first thing you should notice is the perfect s
 \subsection{Improving autoencoder }\label{sec:secondworking}
 {\scriptsize Referenced in: [\ref{sec:imgmaps}] [\ref{sec:netsnext}]  \par}
 
-Given the fairly good AUC score, it looks like to only thing we now need to do, is to increase the size of this autoencoder, and we probably have a really great anomaly detection algorithm. But before we try, and fail\footnote{See chapter \ref{sec:scaling}.}, at this, let us improve our autoencoder first. As you migth agree, the training curve does not look very impressive, and the reconstruction is also not very good. Thats why we suggest some chanced model\footnote{We alter models iteratively, but since we dont want to show tausends of models here, you only see summaries, which is why the changes seem a bit random.}.
+Given the fairly good AUC score, it looks like to only thing we now need to do, is to increase the size of this autoencoder, and we probably have a really great anomaly detection algorithm. But before we try, and fail\footnote{See chapter \ref{sec:scaling}.}, at this, let us improve our autoencoder first. As you migth agree, the training curve does not look very impressive, and the reconstruction is also not very good. Thats why we suggest some changed model\footnote{We alter models iteratively, but since we don`t want to show tausends of models here, you only see summaries, which is why the changes seem a bit random.}.
 
 \begin{figure}[H] 
   \centering
@@ -2062,12 +2065,12 @@ There are some algorithmical changes that we thougth of, that will be testet in
 
 \subsubsection{Physical intuition behind the encoding algorithm}\label{sec:intuitivecode}
 {\scriptsize Referenced in: [\ref{sec:encode0}] [\ref{sec:aultcode}]  \par}
-The usual encoding algorithm could be seen, as inverting a particle decay: Taking for example a simple two particle decay: On the graph, you could understand it as some function, which is making 1 node into two nodes. And as you can find the original particle by some function of the resulting particles, you can use an original particle with some additional attributes\footnote{Like what it decays into (also ignoring uncertainities for now).} to reconstruct the new particles from it. This migth suggest that this kind of autoencoder is optimal for particle physics, but this setup is even more useful as it does not simply cut away additional information, and the physical problem is actually not that optimal, since the number of particles in each decay does not only have to be constant, but also known before in every compression step. Also particles dont decay in steps: It could well be, that the initial particle decays into two particles, of which only one constinuous to decay further. That beeing said, the optimal encoding algorithm, that we would like to be able to write (appendix \ref{sec:aultcode}), would be solve this, and thus have even more physical intuition. 
+The usual encoding algorithm could be seen, as inverting a particle decay: Taking for example a simple two particle decay: On the graph, you could understand it as some function, which is making 1 node into two nodes. And as you can find the original particle by some function of the resulting particles, you can use an original particle with some additional attributes\footnote{Like what it decays into (also ignoring uncertainities for now).} to reconstruct the new particles from it. This migth suggest that this kind of autoencoder is optimal for particle physics, but this setup is even more useful as it does not simply cut away additional information, and the physical problem is actually not that optimal, since the number of particles in each decay does not only have to be constant, but also known before in every compression step. Also particles don`t decay in steps: It could well be, that the initial particle decays into two particles, of which only one constinuous to decay further. That beeing said, the optimal encoding algorithm, that we would like to be able to write (appendix \ref{sec:aultcode}), would be solve this, and thus have even more physical intuition. 
 
 \subsubsection{better encoding}\label{sec:encoding}
 {\scriptsize Referenced in: [\ref{sec:encode0}]  \par}
 Since writing this much more advanced graph abstraction algorithm, would have taken very much time, let us focus first on a bit more simple better encoding algorithm:
-The current encoding basically completely ignores any graph information. After any compression stage the whole graph has to be relearned, and connections only indirectly\footnote{Through the preciding graph update steps.} affect the corresponding feature vectors. Why not use the graph a bit more? Here we suggest that using a function of the original graph as the compressed graph migth be a good idea: When compressing $n$ vectors, you can see the adjacency matrix as a matrix of matrices, and the only task you need to solve, is how to extract some form of this initial global matrix. This is done here, by applying a function to each submatrix. We try out setting this function to be the mean, the maximum or the minimum of the original connections and compare them with or without rounding each entry to be one or zero to the usual graph compression. With the rounding you can see those options as setting a connection to exist when more original connections exist than dont, when at least one connection exist, or when all connections exist.
+The current encoding basically completely ignores any graph information. After any compression stage the whole graph has to be relearned, and connections only indirectly\footnote{Through the preciding graph update steps.} affect the corresponding feature vectors. Why not use the graph a bit more? Here we suggest that using a function of the original graph as the compressed graph migth be a good idea: When compressing $n$ vectors, you can see the adjacency matrix as a matrix of matrices, and the only task you need to solve, is how to extract some form of this initial global matrix. This is done here, by applying a function to each submatrix. We try out setting this function to be the mean, the maximum or the minimum of the original connections and compare them with or without rounding each entry to be one or zero to the usual graph compression. With the rounding you can see those options as setting a connection to exist when more original connections exist than don`t, when at least one connection exist, or when all connections exist.
 THIS DATA IS STILL IN WORK PARTIALLY
 
 The data we compare it on here, includes all the stuff we implement over the remaining chapters, which is why there is an oneoff auc in those tables(see chapter \ref{sec:secmixed}), and also why the quality is generally worse (see chapter \ref{sec:secinv}).
@@ -2150,12 +2153,12 @@ rounded mean & max & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ \\
 \label{table:encode2}
 \end{table}
 
-Evaluating this test series is not as easy as the last one. In the loss, the comparison network is better than all other layers, excluding rounded means with a min function, but the oneoff auc is worse at this model, and at all other ones. That beeing said, when setting the function to be the mean, the usual auc beats the oneoff one a bit. We choose not to use this, because the higher loss means that worse autoencoder produce these results, and a consistent AUC score, that is not reproduced by the oneOff AUC suggests trivial features, which suggest no generality and invertibility.
+Evaluating this test series is not as easy as the last one. In the loss, the comparison network is better than all other layers, excluding rounded means with a min function, but the oneoff auc is worse at this model, and at all other ones. That beeing said, when setting the function to be the mean, the usual auc beats the oneoff one a bit. We choose not to use this, because the higher loss means that worse autoencoder produce these results, and a consistent AUC score, that is not reproduced by the oneoff AUC suggests trivial features, which suggest no generality and invertibility.
 
 
 \subsubsection{better decoding}\label{sec:decoding}
 {\scriptsize Referenced in: [\ref{sec:decode0}]  \par}
-Also the decoder, does not use the graph structure completely. So we try to replace the abstraction with a constant learnable graph, by an abstraction with a graph that is not constant. The problem here, is that the tensorproduct introduced in \ref{sec:identies} and \ref{sec:gnn} does not work for a product of one graph with multiple graphs. The main difficulty lies in finding out how to work with the nondiagonal terms: Consider again adjacency matrices of adjacency matrices: When each feature vector becomes a vector of feature vectors, also each entry in the adjacency matrix becomes a new matrix. These matrices, multiplied with the original entry would result in a tensorproduct, when the new matrices would always be the same, but this is what we want to chance. Finding now the diagonal matrices can be left to a learnable function of the feature vector, but for the offdiagonal matrices, we have two suggestions: The first, graphlike decompresser, define those matrices as functions of the two corresponding diagonal matrices. Here we compare a product, a sum and those rounded versions and and or not only to the abstraction with a constant graph, but also to the second suggestion: paramlike decompresser: instead of the diagonal matrices beeing functions of a feature vector, every submatrix is a learnable function of its two corresponding original feature vectors.
+Also the decoder, does not use the graph structure completely. So we try to replace the abstraction with a constant learnable graph, by an abstraction with a graph that is not constant. The problem here, is that the tensorproduct introduced in \ref{sec:identies} and \ref{sec:gnn} does not work for a product of one graph with multiple graphs. The main difficulty lies in finding out how to work with the nondiagonal terms: Consider again adjacency matrices of adjacency matrices: When each feature vector becomes a vector of feature vectors, also each entry in the adjacency matrix becomes a new matrix. These matrices, multiplied with the original entry would result in a tensorproduct, when the new matrices would always be the same, but this is what we want to change. Finding now the diagonal matrices can be left to a learnable function of the feature vector, but for the offdiagonal matrices, we have two suggestions: The first, graphlike decompresser, define those matrices as functions of the two corresponding diagonal matrices. Here we compare a product, a sum and those rounded versions and and or not only to the abstraction with a constant graph, but also to the second suggestion: paramlike decompresser: instead of the diagonal matrices beeing functions of a feature vector, every submatrix is a learnable function of its two corresponding original feature vectors.
 
 
 \begin{table}[h!]
@@ -2260,11 +2263,11 @@ This shows basically no difference, so since we like the subgraph actions from a
 
 
 
-From our experience, Graph autoencoder have some clear advantages over classical autoencoder. This does not mean, that they dont have problems(as discussed in chapter \ref{sec:whynotgae}), or even that those benefits usually outweigh the problems, but at least that there migth be situations, in which choosing a graph autoencoder would be a good choice.
+From our experience, Graph autoencoder have some clear advantages over classical autoencoder. This does not mean, that they don`t have problems(as discussed in chapter \ref{sec:whynotgae}), or even that those benefits usually outweigh the problems, but at least that there migth be situations, in which choosing a graph autoencoder would be a good choice.
 The most obvious situation, in which graph autoencoder should be choosen, is defined by their input data, if it has the form of a graph: That means multiple vectors, with some relational information between them, that should not be ignored. This can also be beneficial for variable number of vectors, or when a permutation symmetry between the inputs is expected.
 Another benefit is the seperation in multiple similarly handled vectors. This similar handling does not only keep the number of trainable parameters low, and thus makes overfitting hard\footnote{Or in the models we trained basically impossible.}, but also makes interpreting the output easier, since when every attribute of the same kind is treated the same, there are not many differences between the qualities of different particles, but more between different attributes.
 Also and probably most useful, these shared parameters\footnote{The low count and the demanded similarity.} keep the number of needed training samples quite low: Even though more training sample cannot really hurt the network quality, we could without problems, reduce the trainingsize down by more than two orders of magnitude from 600k to 5k (see appendix \ref{sec:asize}), and it seems to be possible to reduce  these training size oven further, allowing us to build useful networks with only $O\left(1\right)$ training samples (see chapter \ref{sec:feyn}).
-Finally, chancing the graph setup layer can chance the whole meaning of a graph layer, transforming a layer that handles physical distance into one that cares only about momenta. This can allow for variable metrik setups, that can iteratively focus on whatever is important at the current position.
+Finally, changing the graph setup layer can change the whole meaning of a graph layer, transforming a layer that handles physical distance into one that cares only about momenta. This can allow for variable metrik setups, that can iteratively focus on whatever is important at the current position.
 
 
 
@@ -2280,11 +2283,11 @@ Given the reasons for why to use graph autoencoder in chapter \ref{sec:whygae},
 \subsubsection{Reproduding vs classifing quality}\label{sec:nogaequal}
 
 When we started working on anomaly detection, Autoencoder seemed like are quite a good idea, a simple way to differentiate between things that are known, and things that are not known, while still giving you a way of testing how good your models are trained, without needing anything else but background data by just evalutating the quality of the autoencoder. It stands to reason that a bad autoencoder did not understand the background data in any way, that can be used to differentiate it from the signal data, but this is basically just an assumption, and so after having a bit more experience encoding and classifying data, and especcially when we now have another method of seperating data, we first want to spend some time evaluating this hypothesis.
-To do this, lets look first at the loss of the autoencoder: Since it is basically just the difference between input and output\footnote{Simplifying chapter \ref{sec:losses} a bit.}, it is a measurement about how good the autoencoder reproduces whatever we put into it\footnote{This is also a bit of a simplification, since the normalization of the data matters a lot, but we will come back to this later.}, and so by just calculating the loss on background data, we have a measure for the quality of the autoencoder.  So basically we want to see a strong falling correlation between the loss of our network and the AUC score\footnote{The lower the loss, the higher the AUC, since we train on QCD jets.}, do we see this? We show here the loss in training against the decision quality of the corresponding network in this trainingstep. Please note that this is exactly what we want, since the correlation we want here, would mean, that in training, a classifier gets continously better\footnote{Yes you could argue, that it is enough when the highest AUC is reached at the lowest loss, but in practice this is not enough because the network does not always reach the same point.}\footnote{It should be noted, that a network cancels the training when it does not improve for a certain number of epochs, so in theory you do not know if there may not even be a better classifier at a way later epoch, but with worse loss. This is usually assumed to be false, since overfitting defines the rest of training, but since we do not see basically any overfitting, this migth not be so easily ignored.}\footnote{We should also note, that this is an analysis that we did only for a small fraction of all networks, since evalutating the quality of hunderts of networks takes considerable time, often even more than training itself. This is also why we chance this method later, to consider only those epochs in which the network improves its loss.}.
+To do this, lets look first at the loss of the autoencoder: Since it is basically just the difference between input and output\footnote{Simplifying chapter \ref{sec:losses} a bit.}, it is a measurement about how good the autoencoder reproduces whatever we put into it\footnote{This is also a bit of a simplification, since the normalization of the data matters a lot, but we will come back to this later.}, and so by just calculating the loss on background data, we have a measure for the quality of the autoencoder.  So basically we want to see a strong falling correlation between the loss of our network and the AUC score\footnote{The lower the loss, the higher the AUC, since we train on QCD jets.}, do we see this? We show here the loss in training against the decision quality of the corresponding network in this trainingstep. Please note that this is exactly what we want, since the correlation we want here, would mean, that in training, a classifier gets continously better\footnote{Yes you could argue, that it is enough when the highest AUC is reached at the lowest loss, but in practice this is not enough because the network does not always reach the same point.}\footnote{It should be noted, that a network cancels the training when it does not improve for a certain number of epochs, so in theory you do not know if there may not even be a better classifier at a way later epoch, but with worse loss. This is usually assumed to be false, since overfitting defines the rest of training, but since we do not see basically any overfitting, this migth not be so easily ignored.}\footnote{We should also note, that this is an analysis that we did only for a small fraction of all networks, since evalutating the quality of hunderts of networks takes considerable time, often even more than training itself. This is also why we change this method later, to consider only those epochs in which the network improves its loss.}.
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/none}
-\caption{(FORGOT THIS PLOT, AND CALCULATION TAKES TIME)A simple old AUC by epoch plot for a unnormated network with thus focus on angular data}
+\caption{(FORGOT THIS PLOT, AND CALCULATION TAKES TIME)A simple old AUC by epoch plot for a unnormalized network with thus focus on angular data}
 \label{fig:none}
   \end{figure}
 
@@ -2309,7 +2312,7 @@ More interrestingly, please note a peculiarity in the preceding images: As you s
 
 
 It should be noted, that this image is still of the bad kind, focussing mostly on the angular part, but as you see, this image still shows a stricly falling relation, while this time it seams to be exponential in nature\footnote{Something like $c - e^{- d \cdot x}$.}. You could ask yourself what is better? The exponential one migth be limited in its quality, but is also easier to saturate. And this is most likely a feature of our network architecture, since you get this curve when you replace the decoder with a more trivial one. Interrestingly the quality in the linear seeming case cannot be linear, since the AUC has to always be below 1, so you could ask yourself how this curve continous. The obvious first assumption would be that both curves are expoential in nature, but we are not able to saturate the capacities of a nontrivially decoding network.
-As logically as this seems, testing this is a bit of a different story: Since apparently our networks dont reach the neccesary quality to saturise themself, this is hard, if not impossible to test. Which is why give up testing just one autoencoder, either by stopping to test just one network, or by testing other kind of classifiers, and as you will see, both suggest a similar, again falling quality curve for little losses: 
+As logically as this seems, testing this is a bit of a different story: Since apparently our networks don`t reach the neccesary quality to saturise themself, this is hard, if not impossible to test. Which is why give up testing just one autoencoder, either by stopping to test just one network, or by testing other kind of classifiers, and as you will see, both suggest a similar, again falling quality curve for little losses: 
 Lets start with multiple networks, instead of plotting parts of one network, we plot the result of multiple networks:
 \begin{figure}[H] 
   \centering
@@ -2320,7 +2323,7 @@ Lets start with multiple networks, instead of plotting parts of one network, we
 
 
 As you see, this relation is basically the same. Since the problem in this step was reproducability, you have a lot of different random network qualities, but, as already mentioned in chapter \ref{sec:secinv}, there is a strong  relation, that would be linear if the x axis would be linear, and that is growing, since it is trained on top data, and in our definitions, the optimal AUC would now be 0 instead of 1. The more interresting part is now those part at really small loss, that seams to deviate from the relation. These are two network types, that have a more complicated decoder, and as you see: They are definitely way better autoencoder, but are also less good classifier.
-Then consider chapter \ref{sec:secother}, and oneOff networks that show basically the same relation!
+Then consider chapter \ref{sec:secother}, and oneoff networks that show basically the same relation!
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/mabe3}
@@ -2329,8 +2332,8 @@ Then consider chapter \ref{sec:secother}, and oneOff networks that show basicall
   \end{figure}
 
 
-So lets us assume, the quality falls of again at some point, why could that be? One reason migth be, that signal and background are similar in certain features. Consider the following network: you feed it one particle only, but not only the 4 momentum, but also the mass: An autoencoder with the rigth compression size would learn to reconstruct the mass from the momentum 4 vector, probably more than it could ever find patterns in the 4 momentum itself. And if you consider that top quarks decay in a similar manner as QCD quarks, there are certainly similarities that an autoencoder should not focus on. This explains mostly why oneOff networks decrease in quality, since the just focus on one feature, but this is also a problem we can solve with a mixed approach, as shown in chapter \ref{sec:secmixed}. On the other hand, this migth explain why autoencoder this effect less, since they combine features, instead of relying on only one.
-That beeing said, this combination of features migth be the real problem: while talking about feature combination, we assumed that a saturated classifier, is an optimal classifier, but this is not actually the case. Consider the c addition explained in chapter \ref{sec:caddition} and tested in chapter \ref{sec:impro}: Any feature that is less useful has a bigger influence, that has to be compensated by a power 3 in this width, and since the leading pt particle is easier to reconstruct\footnote{Meaning the leading pt particle is less random.} than for example the particle with the 7th highest pt, certain parts of the reconstruction are more or less useful, but the combination makes no difference between particles, so their combination will not be optimal, and thus the combination migth only be saturated with bad combination factors. In fact we can look at this, by looking at the partial networks from chapters \ref{sec:scale} and \ref{sec:scale2}\footnote{We should point out, that this is not entirely the same, since instead of adding particles, we here add bunches of 4 particles each, but we would not expect single particles to add differently than particle bunches, and training a graph on a single particle does not exactly make the best use of their relational setup.}: when we dont add them together with their optimal factors, but with each factor beeing 1
+So lets us assume, the quality falls of again at some point, why could that be? One reason migth be, that signal and background are similar in certain features. Consider the following network: you feed it one particle only, but not only the 4 momentum, but also the mass: An autoencoder with the rigth compression size would learn to reconstruct the mass from the momentum 4 vector, probably more than it could ever find patterns in the 4 momentum itself. And if you consider that top quarks decay in a similar manner as QCD quarks, there are certainly similarities that an autoencoder should not focus on. This explains mostly why oneoff networks decrease in quality, since the just focus on one feature, but this is also a problem we can solve with a mixed approach, as shown in chapter \ref{sec:secmixed}. On the other hand, this migth explain why autoencoder this effect less, since they combine features, instead of relying on only one.
+That beeing said, this combination of features migth be the real problem: while talking about feature combination, we assumed that a saturated classifier, is an optimal classifier, but this is not actually the case. Consider the c addition explained in chapter \ref{sec:caddition} and tested in chapter \ref{sec:impro}: Any feature that is less useful has a bigger influence, that has to be compensated by a power 3 in this width, and since the leading pt particle is easier to reconstruct\footnote{Meaning the leading pt particle is less random.} than for example the particle with the 7th highest pt, certain parts of the reconstruction are more or less useful, but the combination makes no difference between particles, so their combination will not be optimal, and thus the combination migth only be saturated with bad combination factors. In fact we can look at this, by looking at the partial networks from chapters \ref{sec:scale} and \ref{sec:scale2}\footnote{We should point out, that this is not entirely the same, since instead of adding particles, we here add bunches of 4 particles each, but we would not expect single particles to add differently than particle bunches, and training a graph on a single particle does not exactly make the best use of their relational setup.}: when we don`t add them together with their optimal factors, but with each factor beeing 1
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/splitscale}
@@ -2348,7 +2351,7 @@ As a final note in this subchapter: Please consider that oneoff networks should
 
 
 These were quite specific problems to top tagging and anomaly detection, but there are some more general problems working with autoencoders, we want to focus here on two:
-Firstly, it is not trivial to find the best compression size (see chapter \ref{sec:csize}. The lower this size, the more feature migth be compared to trivial values, and the higher it is, the more features migth be reconstructed perfectly and thus contain no classification information\footnote{Here the graph structure actually helps, since permutation invariance makes it nontrivial to get an true identity.}. We choose my compressionsize by observing that a 4 node network is only invertible for a compression size of at least 9\footnote{At least for 4 nodes constructed out of 3 features and flag, so a compression size of 9 means that i use $\frac{3}{flag + 4}$ of the inputsize as compressionsize.}, but this still leaves the question open, how to increase the compression size with the number of nodes. You could argue that the more nodes there are, the more features are found that can be used to compress the input, but you could also argue that the added inputs are more random, and thus allow for less compression. This is why we just leave the fraction constant in the chapters that use bigger networks. You migth ask why we dont just test this again for a higher node count, but this would not only be quite timeintensive, but also only solve one more node size.
+Firstly, it is not trivial to find the best compression size (see chapter \ref{sec:csize}. The lower this size, the more feature migth be compared to trivial values, and the higher it is, the more features migth be reconstructed perfectly and thus contain no classification information\footnote{Here the graph structure actually helps, since permutation invariance makes it nontrivial to get an true identity.}. We choose my compressionsize by observing that a 4 node network is only invertible for a compression size of at least 9\footnote{At least for 4 nodes constructed out of 3 features and flag, so a compression size of 9 means that i use $\frac{3}{flag + 4}$ of the inputsize as compressionsize.}, but this still leaves the question open, how to increase the compression size with the number of nodes. You could argue that the more nodes there are, the more features are found that can be used to compress the input, but you could also argue that the added inputs are more random, and thus allow for less compression. This is why we just leave the fraction constant in the chapters that use bigger networks. You migth ask why we don`t just test this again for a higher node count, but this would not only be quite timeintensive, but also only solve one more node size.
 and also there is this image (circle reconstruction problem compared to oneoffs, STILL WORKING ON)
 
 
@@ -2357,7 +2360,7 @@ and also there is this image (circle reconstruction problem compared to oneoffs,
 \subsection{Other algorithms }\label{sec:other}
 {\scriptsize Referenced in: [\ref{sec:mixedidea}]  \par}
 
-Since oneOff networks seem to have potential, that is just not used that well on jets, you could ask yourself if other classical methods work better. So this chapter serves as an introduction into several of those classical algorithms for finding signal events after training on background events, as well as a reasoning why this is not the case. The field these algorithms belong to, is called one class learning.
+Since oneoff networks seem to have potential, that is just not used that well on jets, you could ask yourself if other classical methods work better. So this chapter serves as an introduction into several of those classical algorithms for finding signal events after training on background events, as well as a reasoning why this is not the case. The field these algorithms belong to, is called one class learning.
 
 \subsubsection{Support vector machines}\label{sec:whatssvm}
 
@@ -2374,7 +2377,7 @@ could not ever be learned. This is a problem, since a shape like this, could be
 
 \subsubsection{k neirest neighbours}\label{sec:whatsnvm}
 
-Another usually quite useful algoritm is also an extension of an supervised task: Given two classes of vectors, you can classify each new point, by looking at the class of the vector that is closest to it, or at the mean of the classes of the $k$ neirest vectors to it. This you can extend to the one class case, by setting the loss of one vector to be the mean of the differences to its $k$ neirest neighbours. Since those known points are only background events, you can expect an abnormal event to have a higher loss, while background events are probably more similar to already known background events. The problem comes from its ability to overfit. This can easily be understood in the supervised case, since single weird background cases can lead to region in which no signal can be detected. You could say that autoencoder focus on the distribution of events, Support vector machines focus on the outliers of their distribution, while k neirest neighbour focusses on the whole volume, which means, it could solve the above distribution\ref{fig:oneclasscircle}, with an quite good AUC of $0.89$ for $k = 1$ and $0.96$ for $k = 100$\footnote{There is some overlay between signal and background in the image, which limits the AUC.}, and in the jet case, this algorithm does better, reaching an AUC of ???, but it is still limited by the curse of dimensionality: One class learning algorithm usually work better on low dimensional inputs than on high dimensional one, and here this can be understood quite easily, since the volume of possible vectors grows exponentially with the dimension, while the number of trainingsamples wont chance to much, making the difference between each of the backgroundevents statistically bigger.
+Another usually quite useful algoritm is also an extension of an supervised task: Given two classes of vectors, you can classify each new point, by looking at the class of the vector that is closest to it, or at the mean of the classes of the $k$ neirest vectors to it. This you can extend to the one class case, by setting the loss of one vector to be the mean of the differences to its $k$ neirest neighbours. Since those known points are only background events, you can expect an abnormal event to have a higher loss, while background events are probably more similar to already known background events. The problem comes from its ability to overfit. This can easily be understood in the supervised case, since single weird background cases can lead to region in which no signal can be detected. You could say that autoencoder focus on the distribution of events, Support vector machines focus on the outliers of their distribution, while k neirest neighbour focusses on the whole volume, which means, it could solve the above distribution\ref{fig:oneclasscircle}, with an quite good AUC of $0.89$ for $k = 1$ and $0.96$ for $k = 100$\footnote{There is some overlay between signal and background in the image, which limits the AUC.}, and in the jet case, this algorithm does better, reaching an AUC of ???, but it is still limited by the curse of dimensionality: One class learning algorithm usually work better on low dimensional inputs than on high dimensional one, and here this can be understood quite easily, since the volume of possible vectors grows exponentially with the dimension, while the number of trainingsamples wont change to much, making the difference between each of the backgroundevents statistically bigger.
 
 \subsubsection{Isolation forests}\label{sec:whatsiforest}
 
@@ -2397,7 +2400,7 @@ MAYBE NOT AT THE RIGTH POS
 
 An optimal autoencoder should be equivalent to the network with the compression size set to the input size. The problem here is, that this trivial model does not neccesarily reproduce its input perfectly. As described in chapter \ref{sec:gnn}, the graph update step is given by
 \begin{equation}f{\left(A_{k}^{i} \cdot n_{j}^{k} \cdot x_{i} + s_{j} \cdot x_{i} \right)}\end{equation}
-and this kind of update step is not always invertible through another step. To see this, let us first ignore the activation function as $f{\left(x \right)} = x$ and let us use a fixed size. Given 3 nodes of 2 features each, let the adjacency matrix and thus the graph be fixed to be\footnote{Please note that i set here the diagonal entries to zero, while in my implementation those are usually one, but this does not really matter, since this is just a chance in learnable parameters. Here this is done to simplify the following calculations.}
+and this kind of update step is not always invertible through another step. To see this, let us first ignore the activation function as $f{\left(x \right)} = x$ and let us use a fixed size. Given 3 nodes of 2 features each, let the adjacency matrix and thus the graph be fixed to be\footnote{Please note that i set here the diagonal entries to zero, while in my implementation those are usually one, but this does not really matter, since this is just a change in learnable parameters. Here this is done to simplify the following calculations.}
 \begin{equation}\left[\begin{matrix}0 & 1 & 0\\1 & 0 & 1\\0 & 1 & 0\end{matrix}\right]\end{equation}
 while the 2x2 matrices are general, we can use the kronecker product to convert them, corresponding to converting the 2 dimensional feature vector in an one dimensional one, into an corresponding matrix that can be multiplied to this new one dimensional feature vector. This matrix will then be given by
 \begin{equation}\left[\begin{matrix}s_{00} & s_{01} & n_{00} & n_{01} & 0 & 0\\s_{10} & s_{11} & n_{10} & n_{11} & 0 & 0\\n_{00} & n_{01} & s_{00} & s_{01} & n_{00} & n_{01}\\n_{10} & n_{11} & s_{10} & s_{11} & n_{10} & n_{11}\\0 & 0 & n_{00} & n_{01} & s_{00} & s_{01}\\0 & 0 & n_{10} & n_{11} & s_{10} & s_{11}\end{matrix}\right]\end{equation}
@@ -2412,7 +2415,7 @@ which can only be solved for
 but since $n$ is given, the matrix cannot be invertible\footnote{You could argue, that $n$ is learnable, but this expects a bit much from the learning algorithm. More explicitely reducing the possible neigbourinteractions into a small subset reduces the possibilities of the learning algorithm and the worth of the graph drastically.}.
 
 You could ask yourself if this is actually a problem, since even though two nonactivated update steps cannot invert themself, but surely a bunch of update steps are invertible together. We also assumed the adjacency matrix to be the same, which does not actually has to be the case. And even if not, since the compression size is not the same as the input size, the problem is anyway different.
-Sadly this is not something that we are able to easily calculate, but what we can do, is test this experimentally. As shown above, any graph update step can be rewritten as an product with a specific matrix. This allows us to create an inverse update step, that is equivalent to the normal one, except for a numerical inverse of the update matrix and train networks using those inverse update steps to decompress our data (You could ask yourself if the update matrix is invertible, and in general it is not (a trivial example migth be no neighbour interaction and a nonivertible self interaction\footnote{Aka a no self interaction.}), but in practice this is a problem that can be controlled: It happens that the function used (tf.linalg.inv) fails, but this is rare, can be controlled by the initialiser of those matrices, and even if it fails, the documentation states that this function migth\footnote{Yes, the documentation is not very precise what happens in such a case.} just return noise instead of showing errors. And considering that having a matrix that is not invertible requires each parameter to be exactly tuned, this can be ignored by the parameters constantly chancing. A bigger problem, and the reason, why we do not use these invertible matrices in each decompression phase are those matrices that are nearly uninvertible (have a determinant very close to zero). Since the determinants of the inverts of those matrices are huge, they can amplify noise and thus confuse the minimizationalgorithm. In practice that means that a network that once has reached a quite low loss, can have quite a bigger loss after a couple more training steps). These networks are now tested in two situations, first compared to a normal , quite good, Network, which results in a less smooth learning curve and/but (I SADLY FORGOT THOSE NETWORKS, SO THIS RESULS APPEAR A BIT LATER). Later we test those networks for the identity case. This means not only a compression size of the same size as the input size \footnote{And also no compression steps. This migth seem trivial, but is actually not, since you could interpret a compression as chanching the distribution of features over the number of particles and the number of features for each particle.}, but also afterwards, networks that compress, but are initialised to do nothing (I SADLY FORGOT THOSE NETWORKS, SO THIS RESULS APPEAR A BIT LATER).
+Sadly this is not something that we are able to easily calculate, but what we can do, is test this experimentally. As shown above, any graph update step can be rewritten as an product with a specific matrix. This allows us to create an inverse update step, that is equivalent to the normal one, except for a numerical inverse of the update matrix and train networks using those inverse update steps to decompress our data (You could ask yourself if the update matrix is invertible, and in general it is not (a trivial example migth be no neighbour interaction and a nonivertible self interaction\footnote{Aka a no self interaction.}), but in practice this is a problem that can be controlled: It happens that the function used (tf.linalg.inv) fails, but this is rare, can be controlled by the initialiser of those matrices, and even if it fails, the documentation states that this function migth\footnote{Yes, the documentation is not very precise what happens in such a case.} just return noise instead of showing errors. And considering that having a matrix that is not invertible requires each parameter to be exactly tuned, this can be ignored by the parameters constantly changing. A bigger problem, and the reason, why we do not use these invertible matrices in each decompression phase are those matrices that are nearly uninvertible (have a determinant very close to zero). Since the determinants of the inverts of those matrices are huge, they can amplify noise and thus confuse the minimizationalgorithm. In practice that means that a network that once has reached a quite low loss, can have quite a bigger loss after a couple more training steps). These networks are now tested in two situations, first compared to a normal , quite good, Network, which results in a less smooth learning curve and/but (I SADLY FORGOT THOSE NETWORKS, SO THIS RESULS APPEAR A BIT LATER). Later we test those networks for the identity case. This means not only a compression size of the same size as the input size \footnote{And also no compression steps. This migth seem trivial, but is actually not, since you could interpret a compression as chanching the distribution of features over the number of particles and the number of features for each particle.}, but also afterwards, networks that compress, but are initialised to do nothing (I SADLY FORGOT THOSE NETWORKS, SO THIS RESULS APPEAR A BIT LATER).
 
 So finally we see those invertibility problems as another kind of loss. Next to allowing only for $n$ out of $m$ features to be used, these $n$ features have to work around the structure of the graph. This means, that comparable autoencoder of a non graph type work better for smaller compression size. This migth seem terrible, but we think it only means, that each compression size for a graph network is equivalent to a smaller one for a nongraph network, and thus, some compression sizes close to the maximum are impossible. Finally, you could even see this as an benefit for graph autoencoder, since choosing the rigth compression size is not a trivial task (see chapter \ref{sec:csize}), and this gives you kind of a regulator, saving you from choosing a too high one. Also please note, that the ability for the autoencoder to reconstruct data, does not imply anything concerning its effectiveness as classifier
 
@@ -2451,7 +2454,7 @@ So finally we see those invertibility problems as another kind of loss. Next to
 
 
 %from file ..\..\write\/data\10anhang\15mnist.py
-\subsection{Using oneOff networks on MNist data }\label{sec:amnist}
+\subsection{Using oneoff networks on MNist data }\label{sec:amnist}
 {\scriptsize Referenced in: [\ref{sec:oomnist}]  \par}
 
 
@@ -2471,7 +2474,7 @@ As explained in chapter \ref{sec:imgsetup}, our topK algorithm, on which all gra
 
 (ENTER understanding)
 
-Another thing, that you can do with this, is chance the input definitions. One obvious choice would be to use a simple 4 vector. This results in the following metrik
+Another thing, that you can do with this, is change the input definitions. One obvious choice would be to use a simple 4 vector. This results in the following metrik
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/none}
@@ -2540,7 +2543,7 @@ when two nodes are of the same distance, which to connect? We simply connect bot
   \end{figure}
 
 
-how to connect to nodes that dont have neighbours open? we solve this by no longer requiring the adjacency matrix to be symmetric, and thus the graph to be directed, but you could think about chancing this, by symmetrizing the matrix, either in a way that requires both or either direction to be connnected
+how to connect to nodes that don`t have neighbours open? we solve this by no longer requiring the adjacency matrix to be symmetric, and thus the graph to be directed, but you could think about changing this, by symmetrizing the matrix, either in a way that requires both or either direction to be connnected
 
 (ENTER RESULTS)
 
@@ -2549,7 +2552,7 @@ how to connect to nodes that dont have neighbours open? we solve this by no long
 {\scriptsize Referenced in: [\ref{sec:quickres1}] [\ref{sec:acomparepnet}] [\ref{sec:molnext}]  \par}
 
 
-Since this is just an appendix, I want to take the time, of starting this chapter with a story: A while ago, I was in a chapel. This does not happen very often, and judging from the chapel, it is also not used very often. One thing you immendiatly notice, is that this chapel did not have any banks, but used a lot of chairs, and since at that moment, I desperatly wanted to think about something else, I became faszinated by those chairs: For some reason, even though these chairs are fairly unordered, we dont think of them as just $n$ chairs, but as $r$ rows of $c$ chairs each. Why? how do we abstract those unordered amount of chairs into need rows and collumns, and do we have an algorithm that can do this for us? Since this is still a thesis about graph autoencoder and not my journal, you migth guess the relation: if you simply draw graph connection between each neighbouring chair, you generate a graph like this
+Since this is just an appendix, I want to take the time, of starting this chapter with a story: A while ago, I was in a chapel. This does not happen very often, and judging from the chapel, it is also not used very often. One thing you immendiatly notice, is that this chapel did not have any banks, but used a lot of chairs, and since at that moment, I desperatly wanted to think about something else, I became faszinated by those chairs: For some reason, even though these chairs are fairly unordered, we don`t think of them as just $n$ chairs, but as $r$ rows of $c$ chairs each. Why? how do we abstract those unordered amount of chairs into need rows and collumns, and do we have an algorithm that can do this for us? Since this is still a thesis about graph autoencoder and not my journal, you migth guess the relation: if you simply draw graph connection between each neighbouring chair, you generate a graph like this
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/none}
@@ -2571,8 +2574,8 @@ So how to solve this? one way would be to connect everything below a fixed dista
 \subsubsection{oneoff math}\label{sec:oomath}
 {\scriptsize Referenced in: [\ref{sec:ooquality}] [\ref{sec:nogaegeneral}] [\ref{sec:impro}]  \par}
 Before we talk about how this works, lets talk about the math behind this idea a bit, especially how this kind of networks should handle multiple kinds of information. To do this, let us consider a simple model: Each feature is build out of two gaussian distributions, the first distribution describes the training/background data, and thus has a mean of 1 and some width $\sigma_{1}$ and the second one describes the signal data, it has a mean of $\mu$ and a width of $\sigma_{2}$. This means the decision quality of this feature can be described by $s = \frac{\operatorname{abs}{\left(\mu - 1 \right)}}{\sqrt{\sigma_{1}^{2} + \sigma_{2}^{2}}}$. The higher $s$ is, the bigger the difference between both peaks, and the better the seperation and thus the higher the auc. This migth remind the reader of the math considered in the chapter about c addition (\ref{sec:caddition}), and it actually concludes, that by considering how to combine two background, the math is exactly same as for c addition:
-Given two distributions, with width $s_{1}$ and $s_{2}$ and mean values of $1$, you can combine both distributions into one distribution with smaller width. This width of the distibution $\frac{c \cdot d_{2} + d_{1}}{c + 1}$ is given by $\frac{\sqrt{c^{2} \cdot s_{2}^{2} + s_{1}^{2}}}{c + 1}$ while the mean is still $1$. This function is exactly the same, as was minimized to find the combination with the best AUC in chapter \ref{sec:caddition}. This migth suggest, that the resulting combination of an oneOff network is the combination with the highest possible AUC. Sadly this is simply not sp easy: The problem are the assumptions made in the chapter about c addition(\ref{sec:caddition}): We set the distance between the background and the signal peak to be constant, which results in the width of the distributions to be the only important thing to consider, when combining two distributions. This is fine, when considering features of similar kind, since you can assume their distributions to be similar, but does not anymore here: And when you assume this distance to be more or less random, the calculation becomes a bit more complicated. In fact you can assume, that any other possible peak could be some sort of signal data, that you want to exclude (see chapter \ref{sec:impro}).
-So consider the following model: Given a background peak around 1 width a certain width $s_{1}$, and an improvement of this peak, being more focussed with a width $s_{2} \leq s_{1}$: Which peak is more probable to seperate a random signal from the background? Since the second peak is less wide, it is less probable for a signal peak to overlap it and thus probably results in a higher AUC score\footnote{To be more precise, you optimize the background peak by improving on the function that generates it, this clearly also changes the signal peak, but does this in a more or less random manner. It could improve the AUC, but could also hurt it. The point is, that this chance is random, and thus on average, optimizing an oneOff network migth be useful. We do this more precisely in chapter \ref{sec:impro}.}
+Given two distributions, with width $s_{1}$ and $s_{2}$ and mean values of $1$, you can combine both distributions into one distribution with smaller width. This width of the distibution $\frac{c \cdot d_{2} + d_{1}}{c + 1}$ is given by $\frac{\sqrt{c^{2} \cdot s_{2}^{2} + s_{1}^{2}}}{c + 1}$ while the mean is still $1$. This function is exactly the same, as was minimized to find the combination with the best AUC in chapter \ref{sec:caddition}. This migth suggest, that the resulting combination of an oneoff network is the combination with the highest possible AUC. Sadly this is simply not sp easy: The problem are the assumptions made in the chapter about c addition(\ref{sec:caddition}): We set the distance between the background and the signal peak to be constant, which results in the width of the distributions to be the only important thing to consider, when combining two distributions. This is fine, when considering features of similar kind, since you can assume their distributions to be similar, but does not anymore here: And when you assume this distance to be more or less random, the calculation becomes a bit more complicated. In fact you can assume, that any other possible peak could be some sort of signal data, that you want to exclude (see chapter \ref{sec:impro}).
+So consider the following model: Given a background peak around 1 width a certain width $s_{1}$, and an improvement of this peak, being more focussed with a width $s_{2} \leq s_{1}$: Which peak is more probable to seperate a random signal from the background? Since the second peak is less wide, it is less probable for a signal peak to overlap it and thus probably results in a higher AUC score\footnote{To be more precise, you optimize the background peak by improving on the function that generates it, this clearly also changes the signal peak, but does this in a more or less random manner. It could improve the AUC, but could also hurt it. The point is, that this change is random, and thus on average, optimizing an oneoff network migth be useful. We do this more precisely in chapter \ref{sec:impro}.}
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/img_doublepeak}
@@ -2581,11 +2584,11 @@ So consider the following model: Given a background peak around 1 width a certai
   \end{figure}
 
 
-This combination of two improving methods, C addition for similar features, and statistical improvement for asimilar features, is why we think, that oneOff networks migth work well, but there are two caveats we need to talk about here: First, this optimization does not help at all, when the distance between the background the signal peak is zero, since when the oneOff network focusses on something that is the same in background and signal, making a distribution less wide, only results in both getting smaller. In practice this becomes only a big problem, when you have trivial inputs, consider the case of the autoencoder discussed at the beginning of the chapter: If you chance the normalization to have a trivial $1$ in on of the other outputs, you lose all decision power of the flag variables. A possible solution to this problem will be discussed in the next chapter \ref{sec:secmixed}.
-Another caveat that should be mentioned, is the loss you use for training an oneOff network. The first choice migth be to just minimise $\left(x - 1\right)^{2}$, but if you try this, you notice that the resulting mean is not $1$ but smaller than $1$. To understand this, consider the following model: given a gaussian peak $i$ with a width $\sigma$ and a mean of $1$, which distribution $i \cdot \mu$, with a constant $\mu$, minimizes $\left(i \cdot \mu - 1\right)^{2}$? This loss can be written\footnote{When approximating the number of training samples as infinite.} as
+This combination of two improving methods, C addition for similar features, and statistical improvement for asimilar features, is why we think, that oneoff networks migth work well, but there are two caveats we need to talk about here: First, this optimization does not help at all, when the distance between the background the signal peak is zero, since when the oneoff network focusses on something that is the same in background and signal, making a distribution less wide, only results in both getting smaller. In practice this becomes only a big problem, when you have trivial inputs, consider the case of the autoencoder discussed at the beginning of the chapter: If you change the normalization to have a trivial $1$ in on of the other outputs, you lose all decision power of the flag variables. A possible solution to this problem will be discussed in the next chapter \ref{sec:secmixed}.
+Another caveat that should be mentioned, is the loss you use for training an oneoff network. The first choice migth be to just minimise $\left(x - 1\right)^{2}$, but if you try this, you notice that the resulting mean is not $1$ but smaller than $1$. To understand this, consider the following model: given a gaussian peak $i$ with a width $\sigma$ and a mean of $1$, which distribution $i \cdot \mu$, with a constant $\mu$, minimizes $\left(i \cdot \mu - 1\right)^{2}$? This loss can be written\footnote{When approximating the number of training samples as infinite.} as
 \begin{equation}\operatorname{gauss}{\left(\left(x - 1\right)^{2},1 - \mu,\mu \cdot \sigma \right)}\end{equation}
 where we write $\operatorname{gauss}{\left(a,\mu,\sigma \right)}$ as the application of an gaussian kernel with mean $\mu$ and width $\sigma$ around $a$.
-This function has the same minima as\footnote{Since the constant part does not chance the minima, and the linear part is zero by symmetry.}
+This function has the same minima as\footnote{Since the constant part does not change the minima, and the linear part is zero by symmetry.}
 \begin{equation}\operatorname{gauss}{\left(x^{2},1 - \mu,\mu \cdot \sigma \right)} = \mu^{2} \cdot s^{2} + \left(1 - \mu\right)^{2}\end{equation}
 This function is minimal at $\mu = \frac{1}{s^{2} + 1}$. Since this value is not always $1$, training on this loss, while comparing the signal peak to 1, does not work. You can fix this, by simply training on a different loss $\left(\operatorname{mean}{\left(x \right)} - 1\right)^{2} + \operatorname{std}^{2}{\left(x \right)}$\footnote{Where $\operatorname{mean}{\left(x \right)}$ returns the mean of the input distribution, while $\operatorname{std}^{2}{\left(x \right)}$ returns its variance.} works quite well, or you can just read just the mean of the output distribution, by subtracting the mean of the background peak, instead of 1. Both methods work with similarly good results, even though their output is not always the same. In the following basically always readjusted means are used.
 
@@ -2593,7 +2596,7 @@ This function is minimal at $\mu = \frac{1}{s^{2} + 1}$. Since this value is not
 \subsection{Self improving oneoff networks }\label{sec:impro}
 {\scriptsize Referenced in: [\ref{sec:evaloow}] [\ref{sec:caddition}] [\ref{sec:simplicity}] [\ref{sec:ooquality}] [\ref{sec:secdata}] [\ref{sec:future}] [\ref{sec:nogaegeneral}] [\ref{sec:oomath}]  \par}
 
-We justified oneOff networks before (see \ref{sec:oomath}) by showing that the factor needed to combine two double gaussian features is the same, when we talk about minimizing the the width of the first peak, or when we use c addition to find the maximum auc value. That migth seem great, but there is one assumption, that we kind of glossed over a bit in the chapter about c addition: The difference between both peaks, is not neccesarily defined by the size of the first peak. So here we want to go into detail about why this is not neccesarily a problem, and what consequences can follow from breaking this assumtion.
+We justified oneoff networks before (see \ref{sec:oomath}) by showing that the factor needed to combine two double gaussian features is the same, when we talk about minimizing the the width of the first peak, or when we use c addition to find the maximum auc value. That migth seem great, but there is one assumption, that we kind of glossed over a bit in the chapter about c addition: The difference between both peaks, is not neccesarily defined by the size of the first peak. So here we want to go into detail about why this is not neccesarily a problem, and what consequences can follow from breaking this assumtion.
 The way we try to understand this, is by looking at width vs auc plots, of different c values combining random gaussian double peaks, that have each a random width between $0$ and $2$, while having fixed means of $0$ and $1$ each\footnote{This is still general, since translation and scale invariance give us 2 degrees of freedom per doublepeak.}. After simulating a lot of random distributions, three nontrivial\footnote{With trivial we mean relations that are defined by at least one doublepeak with an AUC of 1.} classes seem to emerge
 \begin{figure}[H]
 \begin{subfigure}{0.3\textwidth}
@@ -2618,7 +2621,7 @@ The way we try to understand this, is by looking at width vs auc plots, of diffe
 
 
 
-As you see, the first relation is pretty much perfect: the lower the width of the first combined resulting peak, the higher the AUC value is. This is the class we want, and we would get if the assumption would be true. Sadly  this is not the only possible result and the second class is not that optimal: These are distributions of supoptimal combination, where the lowest loss, does not result in the highest AUC, but at least into some value that is close to the expected optimum. This class appears in different levels of accuracy, reaching from distributions, that are nearly indifferable from the optimal case, to some, that are definitely not good. Finally the third class, contains combinations that are completely suboptimal: The optimal AUC value is reached at a basically terrible loss, and by decreasing the loss, the AUC becomes bad again. These are what you migth call traps: a very bad classifier is hidden behind a small initial distribution. You can easily see why this cannot ever be filtered out, by considering the case of a trivial feature, that is just always (for signal and background) $1$: the oneoff network will focus entirely on it, since it can reach a loss that is exactly zero, ignoring every feature that would be better at classification, and thus reach a useless classification score, and by looking only at the background distribution, there is nothing you can do\footnote{Except for using a different algorithm, for example a SVM.}. Please note, that in this case, the autoencoder would actually solve the problem: since the feature is trivial, it will be filtered out, and thus cannot be learned from the oneOff network. Combine this with the fact that, from a quick simulation, this case does not seem to appear to commonly
+As you see, the first relation is pretty much perfect: the lower the width of the first combined resulting peak, the higher the AUC value is. This is the class we want, and we would get if the assumption would be true. Sadly  this is not the only possible result and the second class is not that optimal: These are distributions of supoptimal combination, where the lowest loss, does not result in the highest AUC, but at least into some value that is close to the expected optimum. This class appears in different levels of accuracy, reaching from distributions, that are nearly indifferable from the optimal case, to some, that are definitely not good. Finally the third class, contains combinations that are completely suboptimal: The optimal AUC value is reached at a basically terrible loss, and by decreasing the loss, the AUC becomes bad again. These are what you migth call traps: a very bad classifier is hidden behind a small initial distribution. You can easily see why this cannot ever be filtered out, by considering the case of a trivial feature, that is just always (for signal and background) $1$: the oneoff network will focus entirely on it, since it can reach a loss that is exactly zero, ignoring every feature that would be better at classification, and thus reach a useless classification score, and by looking only at the background distribution, there is nothing you can do\footnote{Except for using a different algorithm, for example a SVM.}. Please note, that in this case, the autoencoder would actually solve the problem: since the feature is trivial, it will be filtered out, and thus cannot be learned from the oneoff network. Combine this with the fact that, from a quick simulation, this case does not seem to appear to commonly
 \begin{table}[h!]
 \centering
 \begin{tabular}{|c | c|} 
@@ -2648,7 +2651,7 @@ terrible & $5$ \\
 
 and you can suggest that this will not be a problem.
 
-But to test this, we have to work on actual data, so this is the loss vs AUC relation for a network trained on the compressed space of top jets, trying to find QCD as signal. Please note, that there is a huge difference to the simple case of optimally adding gaussian double peaks: Firstly, there is an unknown\footnote{Unknown, as we cannot really find out, how many informations the network uses to classify a feature. Since l2 normalised networks usually dont set trained parameters to zero, and any gaussian peak can usually help reduce the width of a peak (Central limit theorem), it is actually reasonable to assume, that the whole compressed space, as a transformation of a here 9 dimensional feature vector, is used.} number of features beeing combined. This wont chance to much, since the quadratic addition of $n$ features can be understood as the quadratic addition of one feature with the quadratic addition of $n - 1$ features, but migth be valuable to keep in mind. Secondly, instead of looking at different values of a $c$ factor, combining the peaks, we only look those c values, the network considers at the end of each epoch: That means, we could overlook an optimal AUC in the middle of an epoch, or even miss a good classifier never considered by the network.
+But to test this, we have to work on actual data, so this is the loss vs AUC relation for a network trained on the compressed space of top jets, trying to find QCD as signal. Please note, that there is a huge difference to the simple case of optimally adding gaussian double peaks: Firstly, there is an unknown\footnote{Unknown, as we cannot really find out, how many informations the network uses to classify a feature. Since l2 normalised networks usually don`t set trained parameters to zero, and any gaussian peak can usually help reduce the width of a peak (Central limit theorem), it is actually reasonable to assume, that the whole compressed space, as a transformation of a here 9 dimensional feature vector, is used.} number of features beeing combined. This wont change to much, since the quadratic addition of $n$ features can be understood as the quadratic addition of one feature with the quadratic addition of $n - 1$ features, but migth be valuable to keep in mind. Secondly, instead of looking at different values of a $c$ factor, combining the peaks, we only look those c values, the network considers at the end of each epoch: That means, we could overlook an optimal AUC in the middle of an epoch, or even miss a good classifier never considered by the network.
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/abeoo}
@@ -2657,7 +2660,7 @@ But to test this, we have to work on actual data, so this is the loss vs AUC rel
   \end{figure}
 
 
-As you see, the AUC seems to fall with the width of the first distribution(this is what we want, since we train on top jets), but not reaching the true optimal value. Comparing this to the theoretical expectation is not that easy: the most obvious reason why this does not matter migth be the inaccuracy of the AUC values: the value migth not be optimal, but is very close to the optimal value, while the AUC seems to fluctuate about more than the expected value. More interresting is the relation at less than optimal AUC values. Sadly there are not that many points here. This correlates to the fact, that oneOff networks gain most of their progress in their first epochs(if not epoch), but the points that we see, seem to fit quite nicely to one, if not two lines, building the tails of the theoretically expected relations for an at least close to optimal case. That beeing said, even ignoring the theoretical expectation, and the possibility of another even better combination, this is a relation that validates our training procedure, suggests that the initial autoencoder works at removing traps, and migth even suggest, that one of the reasons, multiple oneOff networks combined are better than only one, comes from the fact, that multiple distributions reduce the noise in the AUC loss relation, and thus gain statistically better AUC values\footnote{Even though it should be noted, that this cannot be the only effect, since this combination usually results in about an increase of 5\%, while this only seems to account for at most 1\%.}.
+As you see, the AUC seems to fall with the width of the first distribution(this is what we want, since we train on top jets), but not reaching the true optimal value. Comparing this to the theoretical expectation is not that easy: the most obvious reason why this does not matter migth be the inaccuracy of the AUC values: the value migth not be optimal, but is very close to the optimal value, while the AUC seems to fluctuate about more than the expected value. More interresting is the relation at less than optimal AUC values. Sadly there are not that many points here. This correlates to the fact, that oneoff networks gain most of their progress in their first epochs(if not epoch), but the points that we see, seem to fit quite nicely to one, if not two lines, building the tails of the theoretically expected relations for an at least close to optimal case. That beeing said, even ignoring the theoretical expectation, and the possibility of another even better combination, this is a relation that validates our training procedure, suggests that the initial autoencoder works at removing traps, and migth even suggest, that one of the reasons, multiple oneoff networks combined are better than only one, comes from the fact, that multiple distributions reduce the noise in the AUC loss relation, and thus gain statistically better AUC values\footnote{Even though it should be noted, that this cannot be the only effect, since this combination usually results in about an increase of 5\%, while this only seems to account for at most 1\%.}.
 
 
 
@@ -2666,7 +2669,7 @@ As you see, the AUC seems to fall with the width of the first distribution(this
 
 \subsubsection{oneoff outside of physics}\label{sec:oomnist}
 {\scriptsize Referenced in: [\ref{sec:ooquality}]  \par}
-This apparently unused potential led us to try them out on more classical evaluation datasets, and we found a paper(ENTER REFERENCE), that not only works fairly similar to oneOff networks\footnote{They use something called a support vector machine, which is probably most easily described as an algorithm that draws a circle like shape around the known datapoints, and classifies everything inside of the shape to be background, and everything outside to be signal. There main idea is to make the shape to be learnable in a deep way. So the main difference to oneoff networks is the fact that here there is a certain region, with the smallest possible size, optimal to be in for the background events, while in oneoff networks the only values that are optimal are exactly one.}, but also evaluates them quite thourougly on MNIST\footnote{MNIST is a set of handwritten digits, that is often used to test new algoritms \cite{mnist}.}. One algorithm they test their algorithm against is based on autoencoders while another uses GANs, and they constantly outperform them. We test here oneOffs on the following task: Given drawings of the number 7, how well can you detect other numbers. They provide also the results from assuming every other number to be the background, and we provide our results in the appendix \ref{sec:amnist}, but here we focus on 7 for now, since it seems not to easy, while also not beeing to hard of a task. They reach an AUC score of $0.946$ with an error of $0.009$, while oneOffs reach a quality of $0.914$ with and error of $0.018$. You could see this, and think that again, they have potential, but they are definitely worse than the reference paper, but this would ignore one fact: There approach does take the whole datavector to retrieve its loss, while our only takes some part, and by retraining the oneoff network, they do not predict the exact same thing\footnote{On average there is a correlation of about $0.6$ between each retraining.}. This means that it should be easy to combine multiple runs into one good classifier, and the math for this (see chapter \ref{sec:caddition}) is even easier here, since every network could reach the same quality, you can set $c = 1$ and just add each value of $\operatorname{abs}{\left(x - 1 \right)}$ together. If you do this with enough reruns\footnote{We used here 25 runs.}, the AUC converges against a value of $0.981$, beating the comparison paper, and thus showing the true potential of oneoff networks for one class learning!
+This apparently unused potential led us to try them out on more classical evaluation datasets, and we found a paper(ENTER REFERENCE), that not only works fairly similar to oneoff networks\footnote{They use something called a support vector machine, which is probably most easily described as an algorithm that draws a circle like shape around the known datapoints, and classifies everything inside of the shape to be background, and everything outside to be signal. There main idea is to make the shape to be learnable in a deep way. So the main difference to oneoff networks is the fact that here there is a certain region, with the smallest possible size, optimal to be in for the background events, while in oneoff networks the only values that are optimal are exactly one.}, but also evaluates them quite thourougly on MNIST\footnote{MNIST is a set of handwritten digits, that is often used to test new algoritms \cite{mnist}.}. One algorithm they test their algorithm against is based on autoencoders while another uses GANs, and they constantly outperform them. We test here oneoffs on the following task: Given drawings of the number 7, how well can you detect other numbers. They provide also the results from assuming every other number to be the background, and we provide our results in the appendix \ref{sec:amnist}, but here we focus on 7 for now, since it seems not to easy, while also not beeing to hard of a task. They reach an AUC score of $0.946$ with an error of $0.009$, while oneoffs reach a quality of $0.914$ with and error of $0.018$. You could see this, and think that again, they have potential, but they are definitely worse than the reference paper, but this would ignore one fact: There approach does take the whole datavector to retrieve its loss, while our only takes some part, and by retraining the oneoff network, they do not predict the exact same thing\footnote{On average there is a correlation of about $0.6$ between each retraining.}. This means that it should be easy to combine multiple runs into one good classifier, and the math for this (see chapter \ref{sec:caddition}) is even easier here, since every network could reach the same quality, you can set $c = 1$ and just add each value of $\operatorname{abs}{\left(x - 1 \right)}$ together. If you do this with enough reruns\footnote{We used here 25 runs.}, the AUC converges against a value of $0.981$, beating the comparison paper, and thus showing the true potential of oneoff networks for one class learning!
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/examples}
@@ -2677,9 +2680,9 @@ This apparently unused potential led us to try them out on more classical evalua
 
 
 %from file ..\..\write\/data\10anhang\35oometrik
-\subsubsection{Some physical interpretability for oneOff networks}\label{sec:oometrik}
+\subsubsection{Some physical interpretability for oneoff networks}\label{sec:oometrik}
 {\scriptsize Referenced in: [\ref{sec:ooquality}] [\ref{sec:ametrikana}] [\ref{sec:observable}]  \par}
-Another example, why oneOff networks migth be quite useful, comes from our experiments to understand them more . Instead of constructing arbitrary features by utilising deep networks, the algorithm used here only combines input features in a linear way. The data we work on here is provided by cern open data as two lepton events from the 2010 datasets. Momentum 4 vectors of muons\cite{setmuon} as background and of electrons\cite{setelectron} as signals. These 4 vectors are multiplied with a linear metrik, reducing it into one dimension, that is evaluated to minimize $\left(\operatorname{abs}{\left(g_{\mu \nu} \cdot p^{\mu} \cdot p^{\nu} \right)} - 1\right)^{2}$. This results in the network learning the following metrik
+Another example, why oneoff networks migth be quite useful, comes from our experiments to understand them more . Instead of constructing arbitrary features by utilising deep networks, the algorithm used here only combines input features in a linear way. The data we work on here is provided by cern open data as two lepton events from the 2010 datasets. Momentum 4 vectors of muons\cite{setmuon} as background and of electrons\cite{setelectron} as signals. These 4 vectors are multiplied with a linear metrik, reducing it into one dimension, that is evaluated to minimize $\left(\operatorname{abs}{\left(g_{\mu \nu} \cdot p^{\mu} \cdot p^{\nu} \right)} - 1\right)^{2}$. This results in the network learning the following metrik
 \begin{table}[h!]
 \centering
 \begin{tabular}{|c | c c c c|} 
@@ -2699,7 +2702,7 @@ $p_{3}$ & $0.0002$ & $-0.0008$ & $-0.0006$ & $0.4998$ \\
 \hline
 
 \end{tabular}
-\caption{Learned metrik values of oneOff networks trained on muon events}
+\caption{Learned metrik values of oneoff networks trained on muon events}
 \label{table:oneoffmuon}
 \end{table}
 
@@ -2717,18 +2720,18 @@ $1.4198$ & $-1.413$ & $-1.4151$ & $-1.4197$ \\
 \hline
 
 \end{tabular}
-\caption{Learned metrik values for a diagonal metrik oneOff networks trained on muon events}
+\caption{Learned metrik values for a diagonal metrik oneoff networks trained on muon events}
 \label{table:oneoffmuondiag}
 \end{table}
 As you see, this still results in a minkowski metrik like result. This time with a flipped sign, and a different scale, which is just a feature of the implementation. Most importantly, this simplified metrik definition, including less noise, results in a much higher auc value of $0.8007$\footnote{You could ask yourself, why we use muons as background events: This is because the relative uncertainity of each electron mass value is much bigger, since the mass is more than two orders of magnitude smaller. Training a (only diagonal) network like this, still results in a minkowski like metrik ($-0.0058$,$0.0043$,$0.0043$,$0.0058$), but the auc value is way worse reaching only $0.5003$ as the expected mean value has way less physical meaning.}
 
 
 %from file ..\..\write\/data\10anhang\36whyoneoffsfail
-\subsection{How a oneOff network can become noninvertible }\label{sec:oofail}
+\subsection{How a oneoff network can become noninvertible }\label{sec:oofail}
 
 
 
-The easiest model for understanding the oneOff width is something like $\sqrt{\operatorname{abs}^{2}{\left(x - \operatorname{mean}{\left(x \right)} \right)} + \operatorname{std}^{2}{\left(x \right)}}$. And while the means usually match the training data, the standart deviation can be of any size. So training on a dataset and comparing it to another dataset with the same mean and less width, results in a noninvertible network. This is nothing we can do anything about, and an effect that is the same when we talk about autoencoder classifier, and is even less probable here, as we try to minimize the width, making it less probable that there is a distribution with lower width. Still this can happen (you can see the frequency in chapter \ref{sec:cross}), but here we want to mention one effect, that is even worse: antiinvertibility: if trained on a, b has lower loss, and if trained on b, a has lower loss. This is an ultra rare effect, as we have only observed it once (or multiple times if you think of the statistical invertibility of chapter \ref{sec:ldm}), and an effect that cannot happen just with an autoencoder, so how does this happen? In general, an oneoff network should not be able to do this, as if one feature has a certain width in a, and a lower feature in b, you should be able to pick the same feature in b resulting in b finding a more complicated, except for the case in which there is another feature in b with lower width, but this would also mean, that the width of the second feature in a would be bigger than of the first feature, since else it would have been choosen, resulting again in b finding a more complicated. Or in math: given $f_{1}^{b} \leq f_{1}^{a}$ the network is not antiinvertible unless $f_{2}^{b} \leq f_{1}^{b}$, but since $f_{1}^{a} \leq f_{2}^{a}$ it has also to be true that $f_{2}^{b} \leq f_{2}^{a}$ so no network can be antiinvertible\footnote{We simplify here a tiny bit, since you could mix two features, but this does not chance the math.}. That beeing said, since we use autoencoder in the front, it can happen, that a feature of the first autoencoder just does not exist in the second one, thus breaking the logical chain, and making antiinvertible networks possible. We only ever saw a single event doing this, and it was a network working on ldm data (see chapter \ref{sec:ldm}). Ldm data is hard to differentiate at best, making noninvertible networks much more likely (in fact, as seen in chapter \ref{sec:cross}, all noninvertible oneOffs in this thesis are trained on ldm data), and by trying to scale using dense networks, at a node number of 25 we got a antiinvertible network
+The easiest model for understanding the oneoff width is something like $\sqrt{\operatorname{abs}^{2}{\left(x - \operatorname{mean}{\left(x \right)} \right)} + \operatorname{std}^{2}{\left(x \right)}}$. And while the means usually match the training data, the standart deviation can be of any size. So training on a dataset and comparing it to another dataset with the same mean and less width, results in a noninvertible network. This is nothing we can do anything about, and an effect that is the same when we talk about autoencoder classifier, and is even less probable here, as we try to minimize the width, making it less probable that there is a distribution with lower width. Still this can happen (you can see the frequency in chapter \ref{sec:cross}), but here we want to mention one effect, that is even worse: antiinvertibility: if trained on a, b has lower loss, and if trained on b, a has lower loss. This is an ultra rare effect, as we have only observed it once (or multiple times if you think of the statistical invertibility of chapter \ref{sec:ldm}), and an effect that cannot happen just with an autoencoder, so how does this happen? In general, an oneoff network should not be able to do this, as if one feature has a certain width in a, and a lower feature in b, you should be able to pick the same feature in b resulting in b finding a more complicated, except for the case in which there is another feature in b with lower width, but this would also mean, that the width of the second feature in a would be bigger than of the first feature, since else it would have been choosen, resulting again in b finding a more complicated. Or in math: given $f_{1}^{b} \leq f_{1}^{a}$ the network is not antiinvertible unless $f_{2}^{b} \leq f_{1}^{b}$, but since $f_{1}^{a} \leq f_{2}^{a}$ it has also to be true that $f_{2}^{b} \leq f_{2}^{a}$ so no network can be antiinvertible\footnote{We simplify here a tiny bit, since you could mix two features, but this does not change the math.}. That beeing said, since we use autoencoder in the front, it can happen, that a feature of the first autoencoder just does not exist in the second one, thus breaking the logical chain, and making antiinvertible networks possible. We only ever saw a single event doing this, and it was a network working on ldm data (see chapter \ref{sec:ldm}). Ldm data is hard to differentiate at best, making noninvertible networks much more likely (in fact, as seen in chapter \ref{sec:cross}, all noninvertible oneoffs in this thesis are trained on ldm data), and by trying to scale using dense networks, at a node number of 25 we got a antiinvertible network
 
 \begin{figure}[H]
 \begin{subfigure}{0.45\textwidth}
@@ -2741,7 +2744,7 @@ The easiest model for understanding the oneOff width is something like $\sqrt{\o
   \includegraphics[width=.8\linewidth]{../imgs/dsantiinv25}
   \label{fig:drantiinv25_2}
 \end{subfigure}%
-\caption{(doubleroc/sep)Antiinvertibility for ldm data, left using the autoencoder loss, and on rigth using oneOff networks}
+\caption{(doubleroc/sep)Antiinvertibility for ldm data, left using the autoencoder loss, and on rigth using oneoff networks}
 \label{fig:drantiinv25}
 \end{figure}
 
@@ -2758,20 +2761,20 @@ Interrestingly is the seperation quality here much better (even if reversed), as
   \end{figure}
 
 
-and a network with less nodes (we tried 9 and 16, sizes that dont show many zero particles \footnote{Zero padded particles to be precise, for more information see chapter \ref{sec:data}.}) does not show any real difference between both datasets. 36 nodes show a difference, but while not beeing invertible, this dataset is at least not antiinvertible, showing how rare those networks are.
+and a network with less nodes (we tried 9 and 16, sizes that don`t show many zero particles \footnote{Zero padded particles to be precise, for more information see chapter \ref{sec:data}.}) does not show any real difference between both datasets. 36 nodes show a difference, but while not beeing invertible, this dataset is at least not antiinvertible, showing how rare those networks are.
 
 
 
 %from file ..\..\write\/data\10anhang\37oopar
-\subsection{Treating oneOff networks as observables }\label{sec:observable}
+\subsection{Treating oneoff networks as observables }\label{sec:observable}
 
 
-If you use a very simple oneOff networks, on momentum 4 vectors (see appendix \ref{sec:oometrik}) results in this network learning the mass of the particles.
+If you use a very simple oneoff networks, on momentum 4 vectors (see appendix \ref{sec:oometrik}) results in this network learning the mass of the particles.
 This is interresting, since this is the same we would do to compare different particles by their 4 momenta and we now have an algorithm to automatically extract this feature just from data. So it migth be fair to assume that by applying this algorithm to more complicated data, we still extract some feature, and that we can use this feature to find anomalies. But giving a feature, you can do more: you can look at statistical information: if you only produce electrons in you detector, but you measure masses that are on average a bit higher than $500 \cdot keV$, you can conclude that their is something else produced but just electrons. The benefit here is, that you can combine multiple events to get lower uncertainities on the variable you care about, and thus you can easier detect irregularities in your dataset. So when we have an automatic feature extractor, it migth be interresting to see if you can differentiate between datasets using this feature.
-We use here $1000000$ jets, of which $0.01$ are not QCD but top jets\footnote{This is about the most we can do giving our dataset.}. This is enough to reach a significance of $4.6$ sigma on a single oneOff network\footnote{No combination of multiple runs.}. So oneOff features migth be appliable to finding new physics. Most interrestingly this probably be applied to any dataset, and so you could define detectorlevel features to directly compare your data to the expectation, assuming your simulation are good enough.
+We use here $1000000$ jets, of which $0.01$ are not QCD but top jets\footnote{This is about the most we can do giving our dataset.}. This is enough to reach a significance of $4.6$ sigma on a single oneoff network\footnote{No combination of multiple runs.}. So oneoff features migth be appliable to finding new physics. Most interrestingly this probably be applied to any dataset, and so you could define detectorlevel features to directly compare your data to the expectation, assuming your simulation are good enough.
 
 %from file ..\..\write\/data\10anhang\38lpt
-\subsection{Chancing the definition of the transverse momentum }\label{sec:alpt}
+\subsection{changing the definition of the transverse momentum }\label{sec:alpt}
 {\scriptsize Referenced in: [\ref{sec:data}]  \par}
 
 
@@ -2785,7 +2788,7 @@ We use here $1000000$ jets, of which $0.01$ are not QCD but top jets\footnote{Th
 {\scriptsize Referenced in: [\ref{sec:tensorproduct}]  \par}
 
 
-There are multiple different ways of implementing such a layer, a notable one is the one used by particleNet \cite{particleNet}: Their graph connectivity is implemented, by just storing all neighbouring vector to each given vector in a set of vectors, this means, they can implement the update procedure as a function of the original and the neighbour vectors\footnote{This function is actually a bit complicated, involving not only convolutions, but also normalisations between them, and they end by concatting the updated vector to the original one, which is something that is not very useful, when you want to reduce the size of your graph.}. This is not exactly what we do here, mostly since the implementation of the graph as just a corresponding set of neighbourvectors demands for computational reasons that each node is connected to a same number of other nodes, and also requires relearning your graph after each step, which we dont want to force our network to do, as explained in chapter \ref{sec:arelearn}, and also would make this less of a graph autoencoder, and more into an autoencoder with some graph update layers in front of it (which migth also not be a good idea, see chapter \ref{sec:ainfront}), since there is no way to reduce the number of nodes for such an implementation, without completely ignoring the graph structure.
+There are multiple different ways of implementing such a layer, a notable one is the one used by particleNet \cite{particleNet}: Their graph connectivity is implemented, by just storing all neighbouring vector to each given vector in a set of vectors, this means, they can implement the update procedure as a function of the original and the neighbour vectors\footnote{This function is actually a bit complicated, involving not only convolutions, but also normalisations between them, and they end by concatting the updated vector to the original one, which is something that is not very useful, when you want to reduce the size of your graph.}. This is not exactly what we do here, mostly since the implementation of the graph as just a corresponding set of neighbourvectors demands for computational reasons that each node is connected to a same number of other nodes, and also requires relearning your graph after each step, which we don`t want to force our network to do, as explained in chapter \ref{sec:arelearn}, and also would make this less of a graph autoencoder, and more into an autoencoder with some graph update layers in front of it (which migth also not be a good idea, see chapter \ref{sec:ainfront}), since there is no way to reduce the number of nodes for such an implementation, without completely ignoring the graph structure.
 Please note the difference: Since we use an adjacency matrix itself to define the graph(in comparison to calculating some derivative from it), you not only have complete control over the graph, that can be used to shrink the graph structure with the number of feature vectors, but you also allow for an arbitrary number of connections for each node\footnote{This is mostly interresting, since it extends the number of possible compression algorithms: They do not anymore have to satisfy keeping the number of connections constant: The number of possible graphs with $n$ nodes is $2^{n \cdot \left(n - 1\right) / 2}$ (ignoring permutation invariance, self connectivity and directed graphs), for $n = 4$ this results in $64$ possible graphs, of which only $6$ are of this kind. This means that much less compressed graphs are possible, and that finding an algorithm, that can pick only those graphs, is much more complicated (see appendix \ref{sec:atopkwhy} for more).}.
 
 %from file ..\..\write\/data\10anhang\45csize
@@ -2944,8 +2947,8 @@ The sorted network reduces an AUC value (trained on QCD) of $0.6351$ into $0.578
 
 
 
-You migth consider the not sortet curve more clean, but it also does not really improve any further at a fairly early epoch, result in the sorting network reaching a loss of about a factor 3 smaller. Seeing this, sorting seems like a clear choice for us, theoretical concerns dont fair well compared to practical results, and so basically all networks in this thesis are sorted. That beeing said, the original deficits of breaking permutations symmetry, could actually be interpreted to mean the opposite: while it is true, that switching each value, except the sorted one, would not result in the same loss, switching any whole node position, still would result in exactly the same loss. In fact, we can use this, to understand why not sorted networks are so bad: The encoding includes a random\footnote{Actually not random, but you could see it like this, if you only see the initial and final node indices.} node permutation, while the decoding does not, so after the autoencoder, the result is a random permuation of the input features, which then are compared to the still initially ordered input features. That this does not work that well should be clear: So either choose the momentum axis as sorting value\footnote{Which would not be what you would want, since locality in real space is much more important than similar energies, as chapter \ref{sec:ametrikana} shows.} or compare your predictions neirly randomly. In fact, you could argue, that this breaks permutation symmetry, as you impose a defined ordering on your node indices. Finally, if you would want to improve this, you migth look at two things: making the comparison variable learnable, would remove the artificial inflated importance of $lp_{T}$\footnote{But maybe also make the training less stable, and add a less controlable importance to some other mixture of features.} and making the decompression chance the node ordering, best case in a learnable way, would make this whole discussion moot, as the network could converge as good with, as without sorting. That beeing suggested, implementing this is not neccesarily easy, as you would not want any function to apply to all nodes, to make sure you dont break permutation symmetry, which for me looks like you restrict yourself to finding a variable to sort by and to reverse an initial sorting would in general not be easy at all, as the initial sorting could be completely random, but would result here probably in a network sorting the nodes by their transverse momentum, as this is the sorting of the initial data, but this seems to us, as a more complicated implementation of our final sorting layer\footnote{That could actually work less well, not only since it needs more calculcation time, but also since this sorting is done at deconstruction, meaning that later graph update layer wont have an effect on it.}\footnote{You migth also ask yourself if you could not just remove the permutation from the encoding layer, but this is easier said than done: As it is true, that the sorting is generally just done for implementation, but as you combine 4 values into for example two, you could have situations, in which node 0 and 3, as well as node 1 and 2 are combined together into (0,3) and (1,2), and even without sorting, reconstruction this, would result in 0,3,1,2, so you would still either need some kind of permutation in the decompression, or some kind of shortcuts between the layers, that encodes the original position: This would not be bad style, as it could result in the network learning to misuse this information to encode arbitrary information, but would also not be very easy to implement, and migth require a nonpermutation invariant compression and decompression function to work well, which would obviously not be ideal, as keeping permutation invariance is the main reason for this chapter.}.
-So finally: sorting seems to be the rigth choice for us, but a more advanced algorithm, migth still be useful: consider the data from chapter \ref{sec:feyn}: sorting by one parameter is not that useful, when you only have boolean datapoints\footnote{Even though in this chapter no real sorting was used, and you could still work with our approach and multiple sorting layers fairly well, assuming tf.math.top\_k is stable (their documentation does not say so, but the implementation is, but this may chance since also tf.argsort is stable at the time of writing this, but they want to implement a not stable version later to improve the speed of this algorithm).}
+You migth consider the not sortet curve more clean, but it also does not really improve any further at a fairly early epoch, result in the sorting network reaching a loss of about a factor 3 smaller. Seeing this, sorting seems like a clear choice for us, theoretical concerns don`t fair well compared to practical results, and so basically all networks in this thesis are sorted. That beeing said, the original deficits of breaking permutations symmetry, could actually be interpreted to mean the opposite: while it is true, that switching each value, except the sorted one, would not result in the same loss, switching any whole node position, still would result in exactly the same loss. In fact, we can use this, to understand why not sorted networks are so bad: The encoding includes a random\footnote{Actually not random, but you could see it like this, if you only see the initial and final node indices.} node permutation, while the decoding does not, so after the autoencoder, the result is a random permuation of the input features, which then are compared to the still initially ordered input features. That this does not work that well should be clear: So either choose the momentum axis as sorting value\footnote{Which would not be what you would want, since locality in real space is much more important than similar energies, as chapter \ref{sec:ametrikana} shows.} or compare your predictions neirly randomly. In fact, you could argue, that this breaks permutation symmetry, as you impose a defined ordering on your node indices. Finally, if you would want to improve this, you migth look at two things: making the comparison variable learnable, would remove the artificial inflated importance of $lp_{T}$\footnote{But maybe also make the training less stable, and add a less controlable importance to some other mixture of features.} and making the decompression change the node ordering, best case in a learnable way, would make this whole discussion moot, as the network could converge as good with, as without sorting. That beeing suggested, implementing this is not neccesarily easy, as you would not want any function to apply to all nodes, to make sure you don`t break permutation symmetry, which for me looks like you restrict yourself to finding a variable to sort by and to reverse an initial sorting would in general not be easy at all, as the initial sorting could be completely random, but would result here probably in a network sorting the nodes by their transverse momentum, as this is the sorting of the initial data, but this seems to us, as a more complicated implementation of our final sorting layer\footnote{That could actually work less well, not only since it needs more calculcation time, but also since this sorting is done at deconstruction, meaning that later graph update layer wont have an effect on it.}\footnote{You migth also ask yourself if you could not just remove the permutation from the encoding layer, but this is easier said than done: As it is true, that the sorting is generally just done for implementation, but as you combine 4 values into for example two, you could have situations, in which node 0 and 3, as well as node 1 and 2 are combined together into (0,3) and (1,2), and even without sorting, reconstruction this, would result in 0,3,1,2, so you would still either need some kind of permutation in the decompression, or some kind of shortcuts between the layers, that encodes the original position: This would not be bad style, as it could result in the network learning to misuse this information to encode arbitrary information, but would also not be very easy to implement, and migth require a nonpermutation invariant compression and decompression function to work well, which would obviously not be ideal, as keeping permutation invariance is the main reason for this chapter.}.
+So finally: sorting seems to be the rigth choice for us, but a more advanced algorithm, migth still be useful: consider the data from chapter \ref{sec:feyn}: sorting by one parameter is not that useful, when you only have boolean datapoints\footnote{Even though in this chapter no real sorting was used, and you could still work with our approach and multiple sorting layers fairly well, assuming tf.math.top\_k is stable (their documentation does not say so, but the implementation is, but this may change since also tf.argsort is stable at the time of writing this, but they want to implement a not stable version later to improve the speed of this algorithm).}
 
 
 
@@ -3024,7 +3027,7 @@ So even though you migth not be able to induce this from a single test only, Bat
 {\scriptsize Referenced in: [\ref{sec:evaloow}]  \par}
 
 
-I am not a big fan of measuring uncertainities for neural networks, by just repeating the training phase and comparing the output, because as chapter (ENTER LINK) shows, we can reduce this uncertainity neirly arbitrarily long, by just training more carefully. Even though this is the reason why most values in this thesis dont have errors attached to them, here is a simple reproducability study
+I am not a big fan of measuring uncertainities for neural networks, by just repeating the training phase and comparing the output, because as chapter (ENTER LINK) shows, we can reduce this uncertainity neirly arbitrarily long, by just training more carefully. Even though this is the reason why most values in this thesis don`t have errors attached to them, here is a simple reproducability study
 
 \begin{figure}[H] 
   \centering
@@ -3084,7 +3087,7 @@ $10$ & $0$ & $0$ \\
 
 Quick answer: No.
 Long answer: Probably no, but not because the quality is neccesarily worse, just because the number of nans (chapter \ref{sec:nan}) increases a lot, making training for a long time very hard and thus resulting in worse classifiers.
-That beeing said, this still means, that if you could handle the nans, you migth profit from more gtopk layer, but we are not able to test this at the moment, and even though multiple different graphs help interpreting graphs as activations (chapter \ref{sec:agaeactivation}), there is not really any physically useful definition of similarity in angles and momenta, but the angles themself, so chancing the graph setup in the middle of the layers, migth not have any effect at all.
+That beeing said, this still means, that if you could handle the nans, you migth profit from more gtopk layer, but we are not able to test this at the moment, and even though multiple different graphs help interpreting graphs as activations (chapter \ref{sec:agaeactivation}), there is not really any physically useful definition of similarity in angles and momenta, but the angles themself, so changing the graph setup in the middle of the layers, migth not have any effect at all.
 
 
 
@@ -3092,11 +3095,11 @@ That beeing said, this still means, that if you could handle the nans, you migth
 
 
 %from file ..\..\write\/data\10anhang\85sizematters
-\subsection{Trainingsize, and why graph autoencoder dont care about it }\label{sec:asize}
+\subsection{Trainingsize, and why graph autoencoder don`t care about it }\label{sec:asize}
 {\scriptsize Referenced in: [\ref{sec:nobias}] [\ref{sec:ae9}] [\ref{sec:quickres1}] [\ref{sec:whygae}]  \par}
 
 
-Graph networks show some properties that usual network dont. One thing is the apparent independence of the training size, and even when you can easily explain this, as having few parameters in your network, this still migth allow you to train on data that was not usable before (see chapter \ref{sec:secuse})
+Graph networks show some properties that usual network don`t. One thing is the apparent independence of the training size, and even when you can easily explain this, as having few parameters in your network, this still migth allow you to train on data that was not usable before (see chapter \ref{sec:secuse})
 
 \begin{figure}[H]
 \begin{subfigure}{0.3\textwidth}
@@ -3178,7 +3181,7 @@ The corresponding tutorial can be found here (ENTER LINK) and the full code is f
 \subsubsection{Datageneration}\label{sec:netsdata}
 
 Datageneration is often the most timeconsuming part of a new neural network, and it would not be different here. So to save some time, we just generate a sample social network. This allows you to ignore privacy settings<you migth not know everything about every user: you would need to decide how to handle a friend about whom you do not know some critical information>, simplifies the problem a bit\footnote{Since an usual facebook user has a lot of information, and often enough hundrets of friends.}, and allows you to clearly define the anomalous data points. That beeing said, this also means, that we could tweak the data in every possible way to make the results arbitrarily good, which is also why this is the only subchapter that works with self generated data.
-This generated network consists of 5000 randomly generated users with 4 attributes (a constant 1 (flag), $a$:an integer between 1 and 3, $b$:an integer either 0 or 1 as well as a normal distributed value that depends on $a$\footnote{A normal distributed value with mean $0$ and standart deviation $1$ added to $2^{a} / 16$ times another normal distribution with mean $1$ and standart deviation $0.1$. This is done just to have some relation between the elements.}. The corresponding connections are generated the following way: each connection has a probability, that depends on the difference in the person vector\footnote{A factor $e^{- \operatorname{abs}^{2}{\left(x_{i} - x_{j} \right)}}$.} and on the difference in the node index\footnote{Another factor $e^{\left(-0.1\right) \cdot \operatorname{abs}{\left(i - j \right)}}$.}. This means, that more similar persons are connected more closely, and that friends of friends are more probably friends. Now we guess on average 5 connections for each person, with respect to the given probabilities, or 2 for the alternative datapoints. We choose these anomalies, since defining less used accounts as signals allows us later to show a benefit of oneOff networks.
+This generated network consists of 5000 randomly generated users with 4 attributes (a constant 1 (flag), $a$:an integer between 1 and 3, $b$:an integer either 0 or 1 as well as a normal distributed value that depends on $a$\footnote{A normal distributed value with mean $0$ and standart deviation $1$ added to $2^{a} / 16$ times another normal distribution with mean $1$ and standart deviation $0.1$. This is done just to have some relation between the elements.}. The corresponding connections are generated the following way: each connection has a probability, that depends on the difference in the person vector\footnote{A factor $e^{- \operatorname{abs}^{2}{\left(x_{i} - x_{j} \right)}}$.} and on the difference in the node index\footnote{Another factor $e^{\left(-0.1\right) \cdot \operatorname{abs}{\left(i - j \right)}}$.}. This means, that more similar persons are connected more closely, and that friends of friends are more probably friends. Now we guess on average 5 connections for each person, with respect to the given probabilities, or 2 for the alternative datapoints. We choose these anomalies, since defining less used accounts as signals allows us later to show a benefit of oneoff networks.
 Now for each person in this network, we only look at the local surrounding of this person. This is done, by taking only the connection of the friends, or friends of friends of this person into account. This generates a lot smaller graphs, that we can now feed into the autoencoder\footnote{You could ask yourself if this reusing of nodes does not result in a lot of overfitting (by learning the nodes themself), but as you see below, that is not the case, possibly because of the low number of parameters in the graph autoencoder.}, but for simplicity we cut a bit on the size of those new graphs, as we allow for at most 70 nodes\footnote{This still keeps $0.9932$ of all data points.}.
 
 \subsubsection{Training}\label{sec:netstrain}
@@ -3202,7 +3205,7 @@ Much more interestingly, is the loss distribution
   \end{figure}
 
 
-As you see, the reconstruction is not very good, as basically all events have a nonzero loss, but maybe even more interrestingly: there is some difference in the reconstruction of out anomal datapoints. This migth seems like you could use this to seperate datapoints, but there is a difficulty: If you use an autoencoder to seperate datasets, you assume that a dataset which the network never saw, will be reconstructed worse than a dataset that is trained on, but here the opposite is the case: the data that is abnormal is easier reconstructed\footnote{This reminds of of the case of nonnormated nets trained on top jets.}, so any seperation is a bit weird: you could just look at something like $1 - loss$, but since you do not have any reasoning for this this makes the training no longer unsupervised. Also probably only this kind of abnormal data will be reconstructed easier\footnote{This is here probably the case, sine the abnormal data is less complicated, as it contains less nodes, you migth be able to handle this, by defining your loss relative to the number of nodes, but this misses the point a bit, as more easy anomalies can still exist, and we can show that oneOff networks can handle them.}, and by negating the loss, you would not get any useful seperation on other datapoints. So what can we do? Use oneOff networks: In their easiest version, they take the mean of the training peak, and define distance as difference to this peak, which would already solve this problem, and in their deep implementation they migth even improve this further. Anyhow why, this works quite well
+As you see, the reconstruction is not very good, as basically all events have a nonzero loss, but maybe even more interrestingly: there is some difference in the reconstruction of out anomal datapoints. This migth seems like you could use this to seperate datapoints, but there is a difficulty: If you use an autoencoder to seperate datasets, you assume that a dataset which the network never saw, will be reconstructed worse than a dataset that is trained on, but here the opposite is the case: the data that is abnormal is easier reconstructed\footnote{This reminds of of the case of nonnormalized nets trained on top jets.}, so any seperation is a bit weird: you could just look at something like $1 - loss$, but since you do not have any reasoning for this this makes the training no longer unsupervised. Also probably only this kind of abnormal data will be reconstructed easier\footnote{This is here probably the case, sine the abnormal data is less complicated, as it contains less nodes, you migth be able to handle this, by defining your loss relative to the number of nodes, but this misses the point a bit, as more easy anomalies can still exist, and we can show that oneoff networks can handle them.}, and by negating the loss, you would not get any useful seperation on other datapoints. So what can we do? Use oneoff networks: In their easiest version, they take the mean of the training peak, and define distance as difference to this peak, which would already solve this problem, and in their deep implementation they migth even improve this further. Anyhow why, this works quite well
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/oohistNETS}
@@ -3211,7 +3214,7 @@ As you see, the reconstruction is not very good, as basically all events have a
   \end{figure}
 
 
-Please note that we dispence of using any number here, measuring how good the reconstruction is, as we could improve it arbitrarily by chancing the data generation
+Please note that we dispence of using any number here, measuring how good the reconstruction is, as we could improve it arbitrarily by changing the data generation
 
 \subsubsection{Whats next}\label{sec:netsnext}
 
@@ -3273,11 +3276,11 @@ We again use a fairly simple setup, consisting only out of a handful of graph up
 
 
 
-There are two things to note here: first, since the mass can reach order of magnitude of $1000 \cdot g / mol$, and since the difference is squared, this can reach very high loss values at the beginning of the training. Combine this with the fact, that masses are very easy to predict, and this is why you see orders of magnitude of chance in the loss function. As you also see, the chance in loss is different between the compressing network, and the noncompressed version: As both networks require similar times to calculate an training epoch, the compressed version requires more than 100 epochs less to reach a similar result. That beeing said, therefore the noncompressed version reaches a sligthly lower minimal loss of $78$ in comparison to $73$, even though it should be noted, that this difference is tiny compared to initial losses, and the compressing version has a bunch more parameters that migth be able to be tweaked to chance this.
+There are two things to note here: first, since the mass can reach order of magnitude of $1000 \cdot g / mol$, and since the difference is squared, this can reach very high loss values at the beginning of the training. Combine this with the fact, that masses are very easy to predict, and this is why you see orders of magnitude of change in the loss function. As you also see, the change in loss is different between the compressing network, and the noncompressed version: As both networks require similar times to calculate an training epoch, the compressed version requires more than 100 epochs less to reach a similar result. That beeing said, therefore the noncompressed version reaches a sligthly lower minimal loss of $78$ in comparison to $73$, even though it should be noted, that this difference is tiny compared to initial losses, and the compressing version has a bunch more parameters that migth be able to be tweaked to change this.
 
 \subsubsection{Whats next?}\label{sec:molnext}
 
-We dont want to call using a compression layer to pool graph networks generally a good idea, but if you have a network that takes an unbearable time, trying out inserting a compression layer migth be good idea. It migth also be interresting to optimize the hyperparameters of the compression layer, or even to alter the setup by for example using an abstraction layer. Finally, this is tested on a fairly easy setup, and it migth be interresting to use this on a more complicated setup like particleNet.
+We don`t want to call using a compression layer to pool graph networks generally a good idea, but if you have a network that takes an unbearable time, trying out inserting a compression layer migth be good idea. It migth also be interresting to optimize the hyperparameters of the compression layer, or even to alter the setup by for example using an abstraction layer. Finally, this is tested on a fairly easy setup, and it migth be interresting to use this on a more complicated setup like particleNet.
 
 
 
@@ -3288,14 +3291,14 @@ We dont want to call using a compression layer to pool graph networks generally
 \subsection{High level machine learning and feynman diagramms }\label{sec:feyn}
 {\scriptsize Referenced in: [\ref{sec:whygae}] [\ref{sec:asort}]  \par}
 
-Machine learning and anomaly detection is usually only used on low level data. Inputs that are easily generated but timeconsuming for humans to understand. But why not apply machine learning to highly abstracted concepts? You migth ask why one would want this: One result migth be something like a theory evaluation method: If you have a number of predictions, this could classify weirdness in the sense of finding predictions that dont match the rest. In the best case you could also extend theories consistently: you can generate new inputs from existing ones. You could automatically bring structure to your predictions, by looking at the compression space of an autoencoder or you could use this to simplify complicated theories. So why dont we do this? two things come to mind: most theories can not be brougth into vector form, and generating a lot of predictions is quite hard. Luckely both are solved by the graph setup: This graph structure is way more powerful, to the point that artificial intelligence research often encodes knowledge in graphs , and since overfitting has not been a problem at all here, also the low number of training samples should not matter here\footnote{There is a second price you pay, when you train on a few datapoints: Not only becomes overfitting more probable, but you also loose generality, as density fluctuations of the different kind of training samples (where these types of samples are defined by the training itself, which makes them hard to filter out) start to matter more. Sadly we cannot really chance this to much.}
+Machine learning and anomaly detection is usually only used on low level data. Inputs that are easily generated but timeconsuming for humans to understand. But why not apply machine learning to highly abstracted concepts? You migth ask why one would want this: One result migth be something like a theory evaluation method: If you have a number of predictions, this could classify weirdness in the sense of finding predictions that don`t match the rest. In the best case you could also extend theories consistently: you can generate new inputs from existing ones. You could automatically bring structure to your predictions, by looking at the compression space of an autoencoder or you could use this to simplify complicated theories. So why don`t we do this? two things come to mind: most theories can not be brougth into vector form, and generating a lot of predictions is quite hard. Luckely both are solved by the graph setup: This graph structure is way more powerful, to the point that artificial intelligence research often encodes knowledge in graphs , and since overfitting has not been a problem at all here, also the low number of training samples should not matter here\footnote{There is a second price you pay, when you train on a few datapoints: Not only becomes overfitting more probable, but you also loose generality, as density fluctuations of the different kind of training samples (where these types of samples are defined by the training itself, which makes them hard to filter out) start to matter more. Sadly we cannot really change this to much.}
 Now consider feynman diagramms: As they are able to encode particle physics in a finite set of graphs, they are at the same time very high level, while also still providing $O\left(1\right)$ samples, which should be barely enough for us to train on, and finding anomalous feynman diagrams migth actually be an interresting way to solve this thesis initial idea of using graphs to find new physics
 
 The corresponding tutorial can be found here (ENTER LINK) and the full code is found here (ENTER LINK)
 
 \subsubsection{Data generation}\label{sec:feyndata}
 
-Datageneration for feynman diagrams means more converting data, instead of outrigth generating them. The problem is, that all diagrams that you find, are usually given as images, and writing an program to read every image into a diagram is absolutely nontrivial, which is why we just converted those diagramms by hand\footnote{You could actually use a graph neural network for this, build similar to the one from the next chapter \ref{sec:build}.}. That beeing said, you could actually ask yourself, if writing an image like autoencoder to work on those images would not be much less work. And even though we would agree, we think this would also work way worse, as you could not differentiate between an image that just looks like a feynman diagram, and an image that actually represents some physical insigth\footnote{You see this quite clearly in another usecase we were thinking about: Recipes are easily generated by a text based gan, or better texts that look like recipes, but generating recipes that actually taste good is much harder, and you dont really have a way to test this(beside cooking for a long time), as your loss could also just say how much your text looks like a recipe.}. If everything looks like a feynman diagramm, you can easily use the loss to differentiate those two cases, since a chance in loss now definitely represents a better reconstruction in the autoencoder we will train. Also by training on images you could again more probably see overfitting, resulting in higher needed training samples, that we dont have.
+Datageneration for feynman diagrams means more converting data, instead of outrigth generating them. The problem is, that all diagrams that you find, are usually given as images, and writing an program to read every image into a diagram is absolutely nontrivial, which is why we just converted those diagramms by hand\footnote{You could actually use a graph neural network for this, build similar to the one from the next chapter \ref{sec:build}.}. That beeing said, you could actually ask yourself, if writing an image like autoencoder to work on those images would not be much less work. And even though we would agree, we think this would also work way worse, as you could not differentiate between an image that just looks like a feynman diagram, and an image that actually represents some physical insigth\footnote{You see this quite clearly in another usecase we were thinking about: Recipes are easily generated by a text based gan, or better texts that look like recipes, but generating recipes that actually taste good is much harder, and you don`t really have a way to test this(beside cooking for a long time), as your loss could also just say how much your text looks like a recipe.}. If everything looks like a feynman diagramm, you can easily use the loss to differentiate those two cases, since a change in loss now definitely represents a better reconstruction in the autoencoder we will train. Also by training on images you could again more probably see overfitting, resulting in higher needed training samples, that we don`t have.
 We use all diagrams from \cite{diagramms}\footnote{These diagramms are of relatively low order.}, that match our filter of only SM diagrams and at most 9 lines, and represent each diagram in the following way: Each line becomes a node, and each two lines that meet in an edge, are connected. This migth seem counterintuitive at first, as we basically switch nodes and edges, but is actually neccesary, since each edge requires two nodes, and in most usual feynman diagramm this is not given, as input aswell as output lines, only have one edge. Then each line(node) is represented by a 14 dimensional vector, onehot\footnote{Onehot encoding means encoding a number that is smaller than $a$ by a vector of values which element $i$ is $1$ is the number is $i$ and $0$ else.} encoding the particle type (gluon,quark,lepton,muon,Higgs, W Boson, Z Boson, photon, proton,jet), 3 special boolean values encoding anti particles\footnote{For simplicity this variable is always zero for lines that are neither input nor output.}, input lines, output lines and a fourtheenth value that is 1 (similar to flag (see chapter \ref{sec:data})).
 \begin{figure}[H]
 \begin{subfigure}{0.45\textwidth}
@@ -3326,7 +3329,7 @@ We use all diagrams from \cite{diagramms}\footnote{These diagramms are of relati
   \end{figure}
 
 
-Also here a fairly easy setup is used, but instead of the compression algorithm, we use the abstraction one, and the paramlike deconstruction algorithm replaces the classical one, to encode the abstraction of a factor 3 (reducing 9 nodes into 3). Therefore we add 3 parameters, as well as a couple more graph update steps. One thing that migth be important later, is that we dont punish the resulting graph structure directly, even though the paramlike decompression algorithm should make this possible, but only indirectly through the fact that a nonsencical graph structure will worsen the quality of the update step.
+Also here a fairly easy setup is used, but instead of the compression algorithm, we use the abstraction one, and the paramlike deconstruction algorithm replaces the classical one, to encode the abstraction of a factor 3 (reducing 9 nodes into 3). Therefore we add 3 parameters, as well as a couple more graph update steps. One thing that migth be important later, is that we don`t punish the resulting graph structure directly, even though the paramlike decompression algorithm should make this possible, but only indirectly through the fact that a nonsencical graph structure will worsen the quality of the update step.
 \begin{figure}[H] 
   \centering
 \includegraphics[width=0.9\textwidth]{../imgs/historyfeyn}
@@ -3434,7 +3437,7 @@ this migth suggest, that weigthing the adjacency matrix directly would be a good
 
 
 As you see, it migth even be a bit to good, as it reconstructs even more 2 input diagramms. (SEARCH FOR ERRORS IN THOSE RECONSTRUCTE DIAS)
-Finally reproducability and the applicability of oneOff networks migth also be interresting here.
+Finally reproducability and the applicability of oneoff networks migth also be interresting here.
 
 
 
@@ -3455,7 +3458,7 @@ The corresponding tutorial can be found here (ENTER LINK) and the full code is f
 
 \subsubsection{Data generation}\label{sec:builddata}
 
-Data is here the biggest problem, as neither can we simply generate building plans ourself, nor is their any database of building plans, that we can use to generate graphs, or at least we did not find any. So basically we dont have any data, until somebody manually translates hundrets of building plans, into any computer readable format, and since we neither have the expertise nor the time to do this, the original idea of using a generative adversial network to generate new buildings is basically impossible. To be able to still give an example of graph generating networks, we replace one of the networks of the GAN by a predefined algorithm. This migth seem, like you can design arbitrary buildings, by just defining what you want, but it has some big problems, because of which we would still suggest the GAN approach more: first, defining what makes a building into a good building is not at all easy. In fact, in the following we just hope for orthogonal walls, by setting the loss to sum over all connections of $\operatorname{abs}{\left(x_{1} \cdot x_{2} \right)}$ in the hope of generating walls that are parallel to the axis, and thus orthogonal to each other. Other than the fact that this loss is really basic, and we would not even expect any nice outputs, even this loss is so complicated, that tensorflow needs about as long to generate the differentials, as it needs to train the whole network, which is also why we choose to reduce the problem to two dimensions. Finally we add some loss to assert that the width of the building is fixed.
+Data is here the biggest problem, as neither can we simply generate building plans ourself, nor is their any database of building plans, that we can use to generate graphs, or at least we did not find any. So basically we don`t have any data, until somebody manually translates hundrets of building plans, into any computer readable format, and since we neither have the expertise nor the time to do this, the original idea of using a generative adversial network to generate new buildings is basically impossible. To be able to still give an example of graph generating networks, we replace one of the networks of the GAN by a predefined algorithm. This migth seem, like you can design arbitrary buildings, by just defining what you want, but it has some big problems, because of which we would still suggest the GAN approach more: first, defining what makes a building into a good building is not at all easy. In fact, in the following we just hope for orthogonal walls, by setting the loss to sum over all connections of $\operatorname{abs}{\left(x_{1} \cdot x_{2} \right)}$ in the hope of generating walls that are parallel to the axis, and thus orthogonal to each other. Other than the fact that this loss is really basic, and we would not even expect any nice outputs, even this loss is so complicated, that tensorflow needs about as long to generate the differentials, as it needs to train the whole network, which is also why we choose to reduce the problem to two dimensions. Finally we add some loss to assert that the width of the building is fixed.
 
 \subsubsection{Training}\label{sec:buildtrain}
 
diff --git a/out/main.toc b/out/main.toc
index cccafd1f99d87d593fe2d1e72a6b051c139d342b..820046ae037047378213b8cc138c83fb38e5093a 100644
--- a/out/main.toc
+++ b/out/main.toc
@@ -28,14 +28,14 @@
 \contentsline {subsubsection}{\numberline {3.6.1}AUC scores}{23}{subsubsection.3.6.1}%
 \contentsline {subsubsection}{\numberline {3.6.2}Losses}{24}{subsubsection.3.6.2}%
 \contentsline {subsubsection}{\numberline {3.6.3}Images}{24}{subsubsection.3.6.3}%
-\contentsline {subsubsection}{\numberline {3.6.4}Oneoff width}{24}{subsubsection.3.6.4}%
+\contentsline {subsubsection}{\numberline {3.6.4}oneoff width}{24}{subsubsection.3.6.4}%
 \contentsline {subsection}{\numberline {3.7}Evaluating the autoencoder }{24}{subsection.3.7}%
 \contentsline {subsubsection}{\numberline {3.7.1}4 nodes}{25}{subsubsection.3.7.1}%
 \contentsline {subsubsection}{\numberline {3.7.2}9 nodes}{25}{subsubsection.3.7.2}%
 \contentsline {subsection}{\numberline {3.8}Evaluating the classifier }{26}{subsection.3.8}%
 \contentsline {subsubsection}{\numberline {3.8.1}4 nodes}{26}{subsubsection.3.8.1}%
 \contentsline {subsubsection}{\numberline {3.8.2}9 nodes}{27}{subsubsection.3.8.2}%
-\contentsline {section}{\numberline {4}Problems}{29}{section.4}%
+\contentsline {section}{\numberline {4}Open questions}{29}{section.4}%
 \contentsline {subsection}{\numberline {4.1}Scaling the network size }{29}{subsection.4.1}%
 \contentsline {subsubsection}{\numberline {4.1.1}Problems in scaling}{29}{subsubsection.4.1.1}%
 \contentsline {subsubsection}{\numberline {4.1.2}Scaling through batches}{29}{subsubsection.4.1.2}%
@@ -45,28 +45,28 @@
 \contentsline {subsection}{\numberline {4.2}Simplicity and invertibility }{34}{subsection.4.2}%
 \contentsline {subsubsection}{\numberline {4.2.1}Simplicity}{34}{subsubsection.4.2.1}%
 \contentsline {subsubsection}{\numberline {4.2.2}Invertibility}{37}{subsubsection.4.2.2}%
-\contentsline {section}{\numberline {5}Solution 1: normalization}{39}{section.5}%
+\contentsline {section}{\numberline {5}Normalization}{39}{section.5}%
 \contentsline {subsection}{\numberline {5.1}Solving invertibility through normalization }{39}{subsection.5.1}%
 \contentsline {subsubsection}{\numberline {5.1.1}The meaning of complexity}{39}{subsubsection.5.1.1}%
 \contentsline {subsubsection}{\numberline {5.1.2}How to normalise an autoencoder}{39}{subsubsection.5.1.2}%
 \contentsline {subsubsection}{\numberline {5.1.3}Using this normalization}{42}{subsubsection.5.1.3}%
-\contentsline {subsubsection}{\numberline {5.1.4}Improving the AUC scores for normated networks}{45}{subsubsection.5.1.4}%
-\contentsline {subsubsection}{\numberline {5.1.5}Scaling in normated networks}{45}{subsubsection.5.1.5}%
+\contentsline {subsubsection}{\numberline {5.1.4}Improving the AUC scores for normalized networks}{45}{subsubsection.5.1.4}%
+\contentsline {subsubsection}{\numberline {5.1.5}Scaling in normalized networks}{45}{subsubsection.5.1.5}%
 \contentsline {subsubsection}{\numberline {5.1.6}Improving the normalization even further}{45}{subsubsection.5.1.6}%
-\contentsline {section}{\numberline {6}Solution 2: Mixed networks}{48}{section.6}%
-\contentsline {subsection}{\numberline {6.1}Oneoff networks }{48}{subsection.6.1}%
+\contentsline {section}{\numberline {6}Mixed networks}{48}{section.6}%
+\contentsline {subsection}{\numberline {6.1}oneoff networks }{48}{subsection.6.1}%
 \contentsline {subsubsection}{\numberline {6.1.1}oneoff quality}{49}{subsubsection.6.1.1}%
-\contentsline {subsection}{\numberline {6.2}Compressed oneOff learning }{50}{subsection.6.2}%
+\contentsline {subsection}{\numberline {6.2}Compressed oneoff learning }{50}{subsection.6.2}%
 \contentsline {subsection}{\numberline {6.3}A good classifier }{51}{subsection.6.3}%
-\contentsline {subsubsection}{\numberline {6.3.1}trained on QCD}{51}{subsubsection.6.3.1}%
-\contentsline {subsubsection}{\numberline {6.3.2}trained on top}{52}{subsubsection.6.3.2}%
-\contentsline {subsection}{\numberline {6.4}Scale }{52}{subsection.6.4}%
-\contentsline {section}{\numberline {7}Other data}{54}{section.7}%
+\contentsline {subsubsection}{\numberline {6.3.1}Trained on QCD}{51}{subsubsection.6.3.1}%
+\contentsline {subsubsection}{\numberline {6.3.2}Trained on top}{52}{subsubsection.6.3.2}%
+\contentsline {subsection}{\numberline {6.4}Scaling for oneoff networks }{52}{subsection.6.4}%
+\contentsline {section}{\numberline {7}Applying this model to other datasets}{54}{section.7}%
 \contentsline {subsection}{\numberline {7.1}Ligth dark matter }{54}{subsection.7.1}%
 \contentsline {subsection}{\numberline {7.2}Other datasets }{58}{subsection.7.2}%
-\contentsline {subsubsection}{\numberline {7.2.1}Quark v gluon}{58}{subsubsection.7.2.1}%
-\contentsline {subsubsection}{\numberline {7.2.2}leptons}{58}{subsubsection.7.2.2}%
-\contentsline {subsection}{\numberline {7.3}Even more comparison }{59}{subsection.7.3}%
+\contentsline {subsubsection}{\numberline {7.2.1}Quark or gluon}{58}{subsubsection.7.2.1}%
+\contentsline {subsubsection}{\numberline {7.2.2}Leptons}{58}{subsubsection.7.2.2}%
+\contentsline {subsection}{\numberline {7.3}Cross comparisons }{59}{subsection.7.3}%
 \contentsline {section}{\numberline {8}Conclusion}{62}{section.8}%
 \contentsline {subsection}{\numberline {8.1}Whats next? }{62}{subsection.8.1}%
 \contentsline {section}{\numberline {9}appendix}{64}{section.9}%
@@ -104,7 +104,7 @@
 \contentsline {subsection}{\numberline {9.10}Building identities out of graphs }{89}{subsection.9.10}%
 \contentsline {subsection}{\numberline {9.11}changing the input feature space }{91}{subsection.9.11}%
 \contentsline {subsection}{\numberline {9.12}Why c addition migth not be perfect }{91}{subsection.9.12}%
-\contentsline {subsection}{\numberline {9.13}Using oneOff networks on MNist data }{91}{subsection.9.13}%
+\contentsline {subsection}{\numberline {9.13}Using oneoff networks on MNist data }{91}{subsection.9.13}%
 \contentsline {subsection}{\numberline {9.14}Metrik analysis }{91}{subsection.9.14}%
 \contentsline {subsection}{\numberline {9.15}The compression algorithm that we would wish to have }{93}{subsection.9.15}%
 \contentsline {subsection}{\numberline {9.16}topK and better graphs }{94}{subsection.9.16}%
@@ -113,10 +113,10 @@
 \contentsline {subsubsection}{\numberline {9.16.3}oneoff math}{97}{subsubsection.9.16.3}%
 \contentsline {subsection}{\numberline {9.17}Self improving oneoff networks }{99}{subsection.9.17}%
 \contentsline {subsubsection}{\numberline {9.17.1}oneoff outside of physics}{101}{subsubsection.9.17.1}%
-\contentsline {subsubsection}{\numberline {9.17.2}Some physical interpretability for oneOff networks}{102}{subsubsection.9.17.2}%
-\contentsline {subsection}{\numberline {9.18}How a oneOff network can become noninvertible }{103}{subsection.9.18}%
-\contentsline {subsection}{\numberline {9.19}Treating oneOff networks as observables }{105}{subsection.9.19}%
-\contentsline {subsection}{\numberline {9.20}Chancing the definition of the transverse momentum }{106}{subsection.9.20}%
+\contentsline {subsubsection}{\numberline {9.17.2}Some physical interpretability for oneoff networks}{102}{subsubsection.9.17.2}%
+\contentsline {subsection}{\numberline {9.18}How a oneoff network can become noninvertible }{103}{subsection.9.18}%
+\contentsline {subsection}{\numberline {9.19}Treating oneoff networks as observables }{105}{subsection.9.19}%
+\contentsline {subsection}{\numberline {9.20}changing the definition of the transverse momentum }{106}{subsection.9.20}%
 \contentsline {subsection}{\numberline {9.21}Model images appliable for reimplementation generated by keras }{106}{subsection.9.21}%
 \contentsline {subsection}{\numberline {9.22}Comparing our graph update layer to particleNet }{106}{subsection.9.22}%
 \contentsline {subsection}{\numberline {9.23}Variating the compression size }{106}{subsection.9.23}%
@@ -130,7 +130,7 @@
 \contentsline {subsection}{\numberline {9.28}The usage of a batchNormalization layer in the middle of the gae }{111}{subsection.9.28}%
 \contentsline {subsection}{\numberline {9.29}Uncertainity and reproducability of AUC values }{113}{subsection.9.29}%
 \contentsline {subsection}{\numberline {9.30}Is it a good idea to relearn the graph at each step }{114}{subsection.9.30}%
-\contentsline {subsection}{\numberline {9.31}Trainingsize, and why graph autoencoder dont care about it }{114}{subsection.9.31}%
+\contentsline {subsection}{\numberline {9.31}Trainingsize, and why graph autoencoder don`t care about it }{114}{subsection.9.31}%
 \contentsline {subsection}{\numberline {9.32}Graph autoencoder as autoencoder with some graph layers in front }{115}{subsection.9.32}%
 \contentsline {subsection}{\numberline {9.33}Why autoencoder reproduce mean values }{115}{subsection.9.33}%
 \contentsline {section}{\numberline {10}appendix 2: Other usecases}{116}{section.10}%