diff --git a/data/01intro/00title b/data/01intro/00title
index eaeecaee040fc219f56d819b51f5e31ddd61ce93..2cc2b9c6f9653088666f4e0908455d7ce8468bf1 100644
--- a/data/01intro/00title
+++ b/data/01intro/00title
@@ -1 +1 @@
-<section Introduction and Literature>
+<section title="Introduction and Literature" label="secintro">
diff --git a/data/01intro/01intro b/data/01intro/01intro
index 50c794396f5a0bf830ba5a484aefd6f76eeffa1b..70343ac411160194abdc1ecbfa856be67e178e54 100644
--- a/data/01intro/01intro
+++ b/data/01intro/01intro
@@ -1,9 +1,27 @@
-<subsection Introduction>
+<subsection title="Introduction" label="intro">
This chapter will be written as the last one
+<ignore>
+<subsubsection TESTING GROUND>
+
+
+
+<table c="4" cap="An examplary testing table" label="test1" mode="full">
+<hline>
+<tline topic 1~#1+x#~#sin(x)#~12>
+<hline>
+<tline a~b~c~d>
+<tline e~f~g~h>
+<hline>
+</table>
+
+
+<reft test1>
+
+</ignore>
Current State of this Thesis
diff --git a/data/01intro/02physics b/data/01intro/02physics
index 795790d54b92eeb89ae888d85624257be9773367..134f96739fa85fae613edbac7173d6aa91874ccd 100644
--- a/data/01intro/02physics
+++ b/data/01intro/02physics
@@ -1,4 +1,4 @@
-<subsection New Physics>
+<subsection title="New Physics" label="physics">
This Chapter is not yet written, since I want to do it justice
diff --git a/data/01intro/03ae b/data/01intro/03ae
index acbfe52d6e03e3a3a87476a3cf64eabdb8fd9bfc..c6f50aa12d83e4525da796c5d7a5bf9e0efb3fa7 100644
--- a/data/01intro/03ae
+++ b/data/01intro/03ae
@@ -1,4 +1,4 @@
-<subsection Autoencoder>
+<subsection title="Autoencoder" label="ae">
Autoencoder are a kind of neuronal Network, in which the function that should be learned is set to the Identity, and to make this learn more than a trivial Function, a compressed state is introduced. This compressed State is of lower Dimension than the Input Space and is generated by a learnable Function called the Encoder from the Input, and serves as Input to another learnable Function, which is called the Decoder. Together, both try to reconstruct the Input. This reconstruction is usually not perfect, since the reduced Dimension does not allow to encode each possible Value perfectly, but it can be quite good, as long as the data contains some patterns. Consider the following 2 dimensional Data:
@@ -6,6 +6,7 @@ Autoencoder are a kind of neuronal Network, in which the function that should be
As you can see, to completely encode the data you would still require 2 dimensions, but you can approximately encode them into 1 dimension quite well, by using one value as encoding and the second one, in the decoder, as a linear function of this<note Since the Number of trainingsamples is finite, you could map every sample into an index, and map those indices again onto the inputs, reaching a zero loss for any Input with an compression size of 1, but not only is finding such a function quite hard for a neuronal Network, it also would not be useful at all, since on any new data, the Network would not work at all. This is why these kind of functions are a part of what is called overfitting for autoencoder>.
+
This combination of a compresser and a decompresser can be quite useful in multiple ways. Ignoring the obvious task of compressing data (see (ENTER REFERENCE) for an example of an autoencoder outperforming a classical compressor), you can give the decompressor noise to generate new versions of an already known kind of data (see (ENTER REFERENCE)), and even though nowadays GANs are used for this Task, autoencoder have still some benefits, allowing for more control over the generated Data. This works, since (in good autoencoders<note This works better in a special way of training an Autoencoder, called a Variational Autoencoder.>) similarity in the compressed Space represent similarity of the Inputs, so not only by changing the compressed Versions of an Input sligthly you can reproduce similar Inputs, but by Identifieng Features in the Input Space you can chance just one attribute of an Input, and you can also merge the Features of two Inputs into one, see for these Applications (ENTER REFERENCE) and (ENTER REFERENCE)
<i f="none" wmode="True">sample combinations of two image Inputs (something like cats and dogs)</i>
That beeing said, the Application that is focussed on in this work, is the Detection of Anomalies. As introduced on the Informatics Side by (ENTER REFERENCE) and a bit later for Particle Physics by (ENTER REFERENCE), 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 (ENTER CHAPTER> of this Autoencoder, to find Events that are not of the same type as the Data the Autoencoder is trained on.
diff --git a/data/01intro/04anomalydetection b/data/01intro/04anomalydetection
index ed203bb4f868e45ed0f73c8902d25b949aec5be8..47226c706dcc8181f9baf0709a95455e1d22d732 100644
--- a/data/01intro/04anomalydetection
+++ b/data/01intro/04anomalydetection
@@ -1,4 +1,4 @@
-<subsection Anomaly Detection>
+<subsection title="Anomaly Detection" label="anoma">
Anomaly Detection, as the Task of finding abnormal Events has a plentitude of Use cases: from improving the Purity of a dataset (ENTER REFERENCE) to fraud and fault detection (see (ENTER REFERENCE) and (ENTER REFERENCE)).
To achieve this, there is a multitude of algorithms, that will be introduced and applied in Chapter (ENTER CHAPTER), which have Applications for example in motor failure detection and nuclear safety. Problems, that can be of onknown form and in which there are few Examples of Interest, that have to be found with high accuracy, not to different from the Task of finding new Physics in an abundance of noise.
diff --git a/data/01intro/05graphs b/data/01intro/05graphs
index 0e5060165ab209ae6efa84f8982d66281fcdc70e..5976cc3cacadf5f7543100d738f92d36a3deff1b 100644
--- a/data/01intro/05graphs
+++ b/data/01intro/05graphs
@@ -1,4 +1,4 @@
-<subsection Graphs>
+<subsection title="Graphs" label="graphs">
A Graph is a mathematical and informatical Concept, that allows you to store a more general Form of Data then just those encoded in Vectors. Namely, Graphs allows storing relational Information of an unbounded<note For computional Reasons, Graphs are not completely unbounded in the following Chapters, but have a maximum size> amount of 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 Vectors, but for practical Application this is quite useful>, and Edges that are pairs of connected Node Indices, and thus encode the relation between those Nodes.
Now there is a plentitude of extensions for this simple graph, for example there 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#.Also there are weighted graphs, in which each Edge gains an additional Value, that encodes how strong the connection between two nodes is.
diff --git a/data/01intro/06graphae b/data/01intro/06graphae
index 2a6829e3a35179e7c92638d359d7b3acc39be655..c864342387fe99728782ccc902f1d3434771ae2c 100644
--- a/data/01intro/06graphae
+++ b/data/01intro/06graphae
@@ -1,4 +1,4 @@
-<subsection Graph Autoencoder>
+<subsection title="Graph Autoencoder" label="gae">
The main Problem of ParticleNet for finding new Physics is its supervised Approach. This means, that each new physics model can only be detected, if you train a special Network for it. Not only would this need a lot of Networks, with the corresponding high number of false positives, but this also limits there effectiveness, as you can only find new Physics that has already been suggested. This is why you could ask yourself if you could not combine the Graph Approach of ParticleNet with the Autoencoder Idea of QCDorWhat. This is the main Idea that is tried to Implement in this Thesis, and this Taks is definitely not trivial: Creating something like a Graph Autoencoder has some Problems, namely the fact, that a compression is usually not local<note graph pooling operations are quite common, since the output of a graph network usually has a different format than its input. The way this is usually done, is by applying a function (mean,max for example) to each node. ParticleNet for example uses a GlobalAveragePooling (ENTER REFERENCE?), so it calculates the average over the nodes for each feature. This kind of pooling works quite well, but is sadly not really appliable to autoencoders, since those functions are not really invertible>. That does not mean, there are no Approaches, just that most Autors shy away from any Approach that chances the Graph Size<note see (ENTER REFERENCE) or (ENTER REFERENCE)>. The first Paper you find, by just searching for a Graph Autoencoder, is a Paper by Kipf et Al (ENTER REFERENCE) and a lot of Paper referencing it. The main Problem here is, that Kipf et Al uses one fixed Adjacency Matrix, and thus one equal Graph Setup, for any Input. This allows for neither the learnable meaning of similarity that appearently makes ParticleNet so good, the variable inputsize discussed in (ENTER LINK) and, probably worst, not for any structural difference in any different Jets. Other Approaches come from the Problem of Graph Pooling Operations, meaning the definition of some kind of Layer, that takes a Graph as Input, and returns a smaller Graph in a learnable Manner<note This is not an entirely solved Problem, but would be quite useful, since it allows for hirachical Learning, similar to the use of Pooling Layers in Convolutional Networks>. DiffPool (ENTER REFERENCE) and MinCutPool (ENTER REFERENCE) migth be good examples for this, but Graph U Nets (ENTER REFERENCE) stand out, since it also gives an Implementation to a anti Pooling layer, and thus allows for a Graph Autoencoder in the way we require it here , which is why the first Approach we tried is based on their Approach. See for this Chapter (ENTER CHAPTER).<ignore> and even though it does not work very good for us, it still creates the basis for every other Approach.</ignore>
diff --git a/data/02tech/00intro b/data/02tech/00intro
index 91f15edfa75e0ebc5112060646062f70eca5074d..78bd7968de1873e61af63c0ad1b7717838897cb9 100644
--- a/data/02tech/00intro
+++ b/data/02tech/00intro
@@ -1 +1 @@
-<section Definitions>
+<section title="Definitions" label="secdef">
diff --git a/data/02tech/01binclass b/data/02tech/01binclass
index 4843901cb61af5cb48909ff0dafbfdd0367a1e25..b5129b01dee81c89426e62a13eb33647ea41652a 100644
--- a/data/02tech/01binclass
+++ b/data/02tech/01binclass
@@ -1,10 +1,10 @@
-<subsection Binary Classification>
+<subsection title="Binary Classification" label="binclass">
The Task of evaluating finding the difference between Background and Signal Data has been studied a lot as a Boolean decision Problem. Notable use cases include (ENTER REFERENCE) and (ENTER REFERENCE). In general, they consider 4 fractions, the fraction of events that are of type background or signal, which are classified either as background and signal.<ignore>, and try to minimize the fraction of events that are classified wrongly.</ignore>
<i f="defttetc" wmode="True">(redoo yourself) Definitions of the 4 Fractions used to evaluate binary classification</i>
-<subsubsection ROC curve>
+<subsubsection title="ROC curve" label="classroc">
For most decision Problems, these Fractions are a functions of some parameter. Consider the following output of an classifier:
<i f="add3" wmode="True">(mmt/add3, prob not perfect)some recqual with decision parameter implemented</i>
Here this Parameter is the Point at which to cut the distribution in a way that everything above will be classified as signal, while everything below is classified as Background. Since the choise of this cut is quite arbitrary, we evaluate every possible Parameter and plot two fractions against each other.
@@ -16,11 +16,11 @@ These two Fractions are plotted against each other, either in a way showing the
or, to focus on the Error Rate
<i f="none" wmode="True">The other way of plotting a roc curve</i>
-<subsubsection Area Under the Curve>
+<subsubsection title="Area Under the Curve" label="classauc">
To simplify comparing ROC scores, you can use an AUC score to summarise it. This AUC score is defined as the integral of the true positive rate over the false positive rate. Obviously this is not perfect, since you reduce a function into only one number, but it is fairly wide accepted, and easy to interpret, as a perfect score would result in an AUC score of 1, while a Classifier that just guesses results in an AUC score of 0.5 and a perfect anticlassifier would result in 0. Also this reduction into only one number can makes the AUC score less errorprone than other values. 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 false negatives). This can result in Networks improving the AUC score by just chancing unimportant parts of the ROC curve.
-<subsubsection other Measures>
+<subsubsection title="other Measures" label="classother">
This Problem of addressed focus is solved by the e30 measure. This measure corresponds to the inverse of the false positive rate corresponding to a true positive rate of 0.3. This means that the e30 score only cares about the accuracy at an acceptable true positive rate, but it also means, that this score is a bit more random, which is why it is not used very often in the following
<ignore>Another Score that is not used is the F1 score (ENTER DELETE?)</ignore>
diff --git a/data/02tech/02evmod b/data/02tech/02evmod
index b3e2b625ca78cd2a8c5c573253ab843a1e79bee1..24352d075cbd6f5f9f46d26304475c0b483d952a 100644
--- a/data/02tech/02evmod
+++ b/data/02tech/02evmod
@@ -1,19 +1,19 @@
-<subsection Problems in Evaluating a Model>
+<subsection title="Problems in Evaluating a Model" label="eval prob">
Even when we can evaluate a Binary Classification Problem<note see Chapter (ENTER CHAPTER)>, this does not mean, that evaluating an Autoencoder is easy. This is a problem, since we basically want to do 2 Things at the same Time, creating an autoencoder and creating a classifier, and there are 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 useless.
-<subsubsection AUC scores>
+<subsubsection title="AUC scores" label="evalauc">
Your first Idea in how to evalutate an Autoencoder, migth be to simply use the Quality of the Classifier (the AUC Score, see Chapter (ENTER CHAPTER)), since the Classifier works 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 (ENTER CHAPTER multi model Plot AUC loss)), but in general this is simply not true, as Chapter (ENTER CHAPTER AUC through zero) shows. 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 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, and to assume that the AUC score correlates.
-<subsubsection Losses>
+<subsubsection title="Losses" label="evalloss">
So why not always use the loss: Look at the Quality of the Autoencoder and try to optimize only it. This again has Problems: Not only requires this still a strong relation between AUC and loss (That is not neccesarily given, consider the Problem of finding the best compression size: The loss will usually 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 (ENTER CHAPTER)>, which makes comparing different Networks only possible, if you neither alter the Loss nor the Normalization.
-<subsubsection Images>
+<subsubsection title="Images" label="evalimg">
This cross comparison Problem, you can easily solve by simply looking at the Images behind the losses<note the jet Image showing Input and Output of the Autoencoder, see for an example (ENTER CHAPTER)>. 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(see Chapter (ENTER CHAPTER)), 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, especcially since what problems you migth see in those images does not neccesarily correspond to what the network fails at. Most notably you seem to focus on angular differences, and mostly neglect differences in #lp_T#.
<i f="none" wmode="True">some image, with nothing learned in pt</i>
Sure, you can also look at the #p_T# reproduction, but this demands weighting importance, and does not make evaluating images any easier.
-<subsubsection Oneoff width>
+<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 Chapters (ENTER CHAPTER) and (ENTER CHAPTER). Because of this, it will be explained in Chapter (ENTER CHAPTER). 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.
diff --git a/data/02tech/04dataprep b/data/02tech/04dataprep
index de6f75e4b02bbcec4df0d3de72eef5b4db2ceda4..2b1eea780787df4f38c7e0d3f9c0ce57dd1c3ac8 100644
--- a/data/02tech/04dataprep
+++ b/data/02tech/04dataprep
@@ -1,4 +1,4 @@
-<subsection Datapreperation>
+<subsection title="Datapreperation" label="data">
Most of the time (always except in Chapter (ENTER CHAPTER)), Jet Data provided by (ENTER REFERENCE) is used, mostly because it is easier to comapre their results. These Jets have a transverse momentum between #550*Gev# and #650*Gev# and a maximum Radius of #LessThan(R,0.8)#.
These Jets take the form of lists containing 200 momentum 4-vectors sortet by their transverse Momentum, so taking only the first #n# vector for each Jet, you a list of the #n# particles carying the most transverse momentum, and thus probably the most important ones. If a jet is defined by less than 200 final particles, those remaining 4 vectors are set to 0.
diff --git a/data/02tech/06explain b/data/02tech/06explain
index 4041101fb16e0b16372a65ba7cbb728485681214..9e5f976c03287d8077f1d918e5b096a3009a9be4 100644
--- a/data/02tech/06explain
+++ b/data/02tech/06explain
@@ -1,17 +1,17 @@
-<subsection Explaining Graphics>
+<subsection title="Explaining Graphics" label="graphics">
-<subsubsection Output Images>
+<subsubsection title="Output Images" label="imgout">
<i f="none" wmode="True">a sample output input image</i>
In those Images, you see each Particles #phi# and #eta#, including any normalisation, plottet once for the input jet in ???, and once for the output jet in ???. This means that a perfect network would mean, that both jets would overlapp. Zero Particles are not shown and there is some indication of the transverse momentum in the size of the dots, which is given proportional to #1+9/(1+lp_T)#<note inverse function, since higher #p_T# result in lower #lp_T#, and some constants to keep the radius finite>. This sadly does allow you to see differences in #lp_T# very well, so you can also look only at #lp_T#. We show those here as a function of the indice
<i f="none" wmode="True">sample lpt image</i>
This way of looking at the Network performance is quite useful for finding patterns in the data. There seem to be Networks that show a high correlation between the angles(see (ENTER CHAPTER)), and it is quite common for the reproduced Values to have less spread than the input one (see (ENTER CHAPTER)). A problem here is, that you can only look at some Images, and finding one good reproduction for each Network is not that hard. To compat this, we use always the same event<note the same event for each training set to be precise>.
-<subsection AUC Feature Maps>
+<subsubsection title="AUC Feature Maps" label="imgmaps">
<i f="none" wmode="True">a sample Featuremap</i>
Since a L2 loss is just a mean over a lot of L2 losses for each Feature and Particle, you could use the original losses, to also get an AUC score for each Feature and Particle. These AUC scores are shown in Feature Maps, showing the Quality of each combination of Feature on the horizontal axis and Particle on the verticle axis in the form of pixels. A perfect classifier(#Eq(AUC,1)#) would result in a dark blue pixel, a perfect anti classifier(#EQ(AUC,0)#) would be represented by a dark red pixel, and a useless classifier, that guesses or always results in the same class(#Eq(AUC,1/2)#) would be a white classifier. In short: The more colorful a pixel is, the better it is, and an Autoencoder trained on qcd events should be blue, while an Autoencoder trained on top events has to be red.
The nice thing about those maps, is that they can show the focus of the Network, as well as its problems. Since a perfect reconstruction has no decicion power, the same as a terrible reconstruction, a Network that has focus problems<note reconstructs some things much better than other things, making both parts worse>, can be clearly seen in those maps (see (ENTER CHAPTER)). Also it is fairly common, to get one feature and Particle, that has more decision power, than the whole combined Network (see (ENTER CHAPTER) for an example and (ENTER CHAPTER) for the explanation). Finally, an AUC map that is completely blue or red is quite uncommon, more probably some Features are red, some are blue, and you get an indication on which Features are useful for the current Task (see (ENTER CHAPTER)).
-<subsection Network Setups>
+<subsubsection title="Network Setups" label="imgsetups">
<i f="none" wmode="True">a sample Network Image, see below or probably just 900</i>
We show the Setup of each Network as Images similar to those<note these Images are not made for you to be able to write the whole Network yourself, but to understand some of the parts that are switched. For a more technical model description, look at the Images keras generates. See for this (ENTER CHAPTER) or in the code at createmodel.py or at encoder.png and decoder.png>. The Information travels from the left side to the rigth side, through a lot of layers, which all have their own symbol. Some symbols are explained at the Point where they are used, but here those that are shown in Image (ENTER IMAGE) from left to rigth
<list>
diff --git a/data/02tech/09graphnet b/data/02tech/09graphnet
index 3def2fb30fe1cc68dfbce3271e015719337494d9..a935fb11956c9b458872aeb0dac688eb35f98005 100644
--- a/data/02tech/09graphnet
+++ b/data/02tech/09graphnet
@@ -1,4 +1,4 @@
-<subsection Graph Neuronal Networks>
+<subsection title="Graph Neuronal Networks" label="gnn">
Graph Neuronal Networks are defined by a Graph Update Layer. This Layer takes all Feature Vectors of some graph, aswell as their corresponding Graph Connections to return an updated Feature Vector. To do this, this Layer is build from two different Interactions, the Update step of each node itself, which is call the self Interaction Term here, and the Update step of a node corresponding to its neighbouring Nodes in the Graph. This is call the neighbour Interaction Term.
There are multiple different ways of Implementing such a Layer, a notable one would be the one used by ParticleNet (ENTER REFERENCE): 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 require relearning your Graph after each step, which migth not be the best Idea, as explained in Chapter (ENTER CHAPTER), 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 (ENTER CHAPTER)), since there is no way to reduce the number of nodes for such an implementation, without completely ignoring the graph structure.
diff --git a/data/03graphae/00title b/data/03graphae/00title
index cbd1b2c120a300028e2bd44014da120b30d87746..18e6780c89073ea68c83f9d59cd21bd229e9194e 100644
--- a/data/03graphae/00title
+++ b/data/03graphae/00title
@@ -1 +1 @@
-<section Making Graph Autoencoder work>
+<section title="Making Graph Autoencoder work" label="secgae">
diff --git a/data/03graphae/01failed b/data/03graphae/01failed
index c92d0f2487937cde6d120414118b8f0995ec3782..df45a59a7b9972764535d4d2936bf00b861f3ddf 100644
--- a/data/03graphae/01failed
+++ b/data/03graphae/01failed
@@ -1,16 +1,16 @@
-<subsection Failed Approaches>
+<subsection title="Failed Approaches" label="failed">
In this Chapter we will quickly go over some Ideas you could have, on how to implement a graph autoencoder and finish with the first implementation that could be considered working.
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 (ENTER CHAPTERLINK).
-<subsubsection trivial models>
+<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 still 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 (ENTER CHAPTER)). 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.
That beeing said, permutation Invariance can also be a benefit, especially in Permutation Invariant Input data, more to this in Chapter (ENTER CHAPTERLINK)
-<subsubsection minimal models>
+<subsubsection title="minimal models" label="failedminimal">
To improve this model, we started working with smaller Graph sizes (mostly the first 4 particles), making the structure less complicated, and allowing for more experimentation thanks to the lower time cost. Notable improvements include replacing the added zeros by a learnable Function of the remaining parameters, relearning the graph on the new parameterspace and adding some Dense Layers after the Graph Interactions, but the most important Improvment was achieved by making the compression and decompression local in some learning axis. Instead of just removing parameters in an arbitrary way of physical intuition, we demand that particles which are similar in some way are to be compressed together: This is achieved by the creation of a Function that compresses a set of particles into one particle, and allow the Network to learn what similarity means<note In the compression step, we define a new Feature for each node, by which we sort the set of nodes, and afterwards we build sets of n particles from this ordering, and compress them using a linear function (it migth be interresting to look at nonlinear functions, but we generally see worse results by adding an alinearity). Please note that since we use a Feature to sort the elements, and in the Graph Update step there are neighbour steps, that generally increase similarity, connected particles are more probably compressed together.>.
These Networks still have problems, as we will discuss in the following, but generally produce respectable Decision Qualities, and show ("sometimes", see Chapter (ENTER CHAPTERLINK)) similarities between Input and Output Image. These Network is discussed in the next subchapter.
diff --git a/data/03graphae/02simpleae b/data/03graphae/02simpleae
index c79e73a6a13949d431b74c74fc652cb577ba3771..bcf71a6abbf07a079f6d0f4abb51e84d07a77d54 100644
--- a/data/03graphae/02simpleae
+++ b/data/03graphae/02simpleae
@@ -1,4 +1,4 @@
-<subsection An explicit look at the first working graph autoencoder>
+<subsection title="An explicit look at the first working graph autoencoder" label="firstworking">
<i f="none" wmode="True">modeldraw of b1/00 (achte auf dense und das batchnorm vor compare ist)</i>
@@ -6,35 +6,35 @@
As you see, this Autoencoder takes qcd jets, transforms them as introduced in (ENTER CHAPTER), does some additional preprocessing, after which a graph is constructed, an graph update step is run, after which a graph compression algorithm is applied just to be reversed afterwards, to reconstruct a new graph, which is again used to update the feature vector, after which (and after a sorter), the current graph is compared. There are some Layers, that have not explicitely been explained before, so this is what we want to do in this Chapter
-<subsubsection topK>
+<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 solved by demanding that the metrik is entirely diagonal, 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 (ENTER APPENDIX)).
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 (ENTER APPENDIX).
Finally, it should be noted, that the topK layer can increase the size of the Feature vectors, which is useful for the compression algorithm, even though in this specific example this is not used.
-<subsubsection Compression>
+<subsubsection title="Compression" label="quickcompression">
Also this Layer is defined by a learnable sense of locality, but instead of defining a metrik, we simply use the last parameter of the input feature vectors<note This is essentially the same, since it is preceded by a graph update layer, which redefines each Feature with its self interaction matrix, but it has one definite benefit, as the neigbour interaction matrix, tends to average graph connected feature vectors, and thus two nodes, that are connected in the adjacency matrix, are more similar> to sort each feature vector, to split the list of feature vectors in lists of feature vectors, that are compressed together<note in this example, the 4 input features are compressed by sorting into sets of 4 feature vectors, so you could ask yourself if it actually does something to sort by a learnable sense of locality for minimal models, but they at least help to keep the permutation invariance>, which are then transformed using a simple dense layer into a new feature vector each.
<i f="none" wmode="True">compression pictograms</i>
After this Graph compression, which reduces #4# times #3+flag# parameters into one vector with #10# parameters, we use 3 dense layers reducing the parameter count down to #6#
-<subsubsection Decompression>
+<subsubsection title="Decompression" label="quickdecompression">
After inverting the Dense Layers, we again have a compressed Vector with 10 Features.
As each list of vectors is compressed into only one feature vector, the inverse procedure transforms each vector into a list of feature vector. Again we use a simple dense layer
<i f="none" wmode="True">decompression pictograms</i>
-<subsubsection Sorting>
+<subsubsection title="Sorting" label="quicksort">
Since the compression stage reordered each feature vector, while the decompression algorithm does not reorder, comparing lists of vectors is not that easy. It should be clear, that a vector that may be perfectly reconstructed, but shuffled in a random way, can have an neirly arbitrarily big loss. This is solved by the initial and final sorting. These layers simply order each feature vector by its #lp_T# Value, so that at least perfectly reconstructed feature vectors are compared the rigth way. It should be noted, that this sorting is not strictly neccesary, as the Network will learn to work even when you just compare vectors in the way, the nodes are given, but sorting makes this task easier, allowing the Network to focus on more important things, and thus generally working better<note The central problem migth be, that the sorting kind of breaks graph permutation symmetry>. See Chapter (ENTER CHAPTER) for an experimental comparison on a more advanced graph autoencoder.
-<subsubsection Training Setup>
+<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="none" 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 (ENTER CHAPTER)), and you could even ask yourself if it would not be possible to remove the whole need if 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 errorbased than usual training curves, make EarlyStopping still a viable training callback, and result in most of the reasons, each training stops.
-<subsubsection Results>
+<subsubsection title="Results" label="quickres1">
So why do we consider this the first working model?
This migth be the first model, that can show some resemblence between the Input and the Output
<i f="none" wmode="True"> b1/00/simpledraw ml, auf cherrypick hinweise</i>
diff --git a/data/03graphae/03goodae b/data/03graphae/03goodae
index 53323d4d96f64e58fab6bee63581cb41ec935b6a..864cbf1d4a292b8f238e5261a98db91ae8036931 100644
--- a/data/03graphae/03goodae
+++ b/data/03graphae/03goodae
@@ -1,13 +1,13 @@
-<subsection Improving Autoencoder>
+<subsection title="Improving Autoencoder" label="secondworking">
This good AUC score, looks like to only thing we now need to do, is to increase the size, and we probably have an Autoencoder that can rival the best other Autoencoder. But before we try, and fail<note see Chapter (ENTER CHAPTER)>, 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.
<i f="none" wmode="True"> network plot for c1/200</i>
-<subsubsection Training Setup>
+<subsubsection title="Training Setup" label="quick2setup">
Here we still train on the first #4# nodes, and with the same batch size of #200#, but with a lower learning rate of #0.0003# and with a higher patience, stopping only after the network do not improve for 30 epochs. We also increase the compression size from 6 to 7.
-<subsubsection Results>
+<subsubsection title="Results" label="quick2results">
<i f="none" wmode="True"> history for c1/200</i>
diff --git a/data/03graphae/03optimalae b/data/03graphae/03optimalae
index ace17e6cbdf880866658b08b4d56116da590cc42..c8944939e5ba9a4e7d30be34f0d6d37c08a53427 100644
--- a/data/03graphae/03optimalae
+++ b/data/03graphae/03optimalae
@@ -1,16 +1,93 @@
-<subsection Making Autoencoder good>
+<subsection title="Making Autoencoder good" label="thirdworking">
-<subsubsection better encoding>
+<subsubsection title="better encoding" label="encoding">
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 adding 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 the global matrix. This is done here, by applying a function to each submatrix. Here, 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.
-(ENTER RESULTS)
+<table caption="Quality differences for different encoder with a learnable handling of the Feature vectors" label="encode1" c=6>
+<hline>
+<tline " ~mean loss~loss std~n~auc~oneoff auc">
+<hline>
+
+<tline Rounded Min~#0.366#~0.037~5~#0.533#~#0.623#>
+<tline Rounded Max~#0.388#~0.042~6~0.563~0.484>
+<tline Rounded Mean~#0.342#~0.036~5~0.560~0.556>
+<hline>
+<tline Min~0.371~0.025~6~0.559~0.562>
+<tline Max~0.372~0.025~6~0.565~0.541>
+<tline Mean~0.351~0.04~5~0.540~0.486>
+
+<hline>
+
+</table>
+
+
+
Also we test, how well a learnable parameter transformation, as used before, works, compared to also applying a function (mean, max, min) to each feature vector
(ENTER RESULTS)
-<subsubsection better 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 (ENTER CHAPTER) 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 learnabl function of its two corresponding original feature vectors.
+<table caption="Quality differences for different encoder with a fixed function" label="encode2" c="7" mode="classic2">
+<hline>
+<tline " ~function~mean loss~loss std~n~auc~oneoff auc">
+<hline>
+
+<tline Rounded Min~mean~0.366~0.016~4~0.656~0.472>
+<tline Rounded Max~mean~0.333~-1~1~0.656~0.549>
+<tline Rounded Mean~mean~0.372~0.016~4~0.656~0.542>
+<hline>
+<tline Min~mean~0.370~0.007~4~0.656~0.503>
+<tline Max~mean~0.362~0.002~3~0.654~0.550>
+<tline Mean~mean~0.363~0.010~4~0.655~0.505>
+<hline>
+<tline Rounded Mean~min~0.296~0.022~5~0.579~0.543>
+<tline Rounded Mean~max~-1~-1~-1~-1~-1>
+
+<hline>
+
+</table>
+<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 (ENTER CHAPTER) 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.
(ENTER RESULTS)
+
+
+<table caption="Quality differences for different graph like decoder" label="decode1" c=6>
+<hline>
+<tline " ~mean loss~loss std~n~auc~oneoff auc">
+<hline>
+
+<tline Product~0.265~0.019~4~0.568~0.557>
+<tline Sum~0.305~0.026~5~0.566~0.514>
+<tline Or~0.404~0.248~18~0.562~0.502>
+<tline And~0.354~0.074~22~0.587~-1>
+
+<hline>
+</table>
+
+
We can also look at the way, the original Graph is combined with the newly generated Graph. Instead of using a product, we can also use a sum, or again round the result to an or<note In Practice we expect the product to be virtually identical to the or, since the inputs are either 1 or 0> or an and. Since the combination with a constant graph is not very interresting, we use paramlike decompression for the practical results:
(ENTER RESULTS)
+<table caption="Quality differences for different param like decoder" label="decode2" c=6>
+<hline>
+<tline " ~mean loss~loss std~n~auc~oneoff auc">
+<hline>
+
+<tline Product~0.280~0.037~4~0.553~0.522>
+<tline Sum~0.300~0.024~4~0.549~0.526>
+<tline Or~-1~-1~10~-1~-1>
+<tline And~0.348~0.038~16~0.631~0.546>
+
+<hline>
+</table>
Finally, since we now have a little Adjacency Matrix, and a list of Feature vectors for each original Feature vector, we can apply a Graph Update Step on those subgraphs, to hopefully enhance the decompression by mixing the decompression of the Adjacency matrix and the Feature vectors. This was already enabled in all previous compression and decompression tests, but is tested here again on paramlike decompression:
(ENTER RESULTS)
+<table caption="Quality difference for either running learnable sub graph updates or not" label="decode3" c=6>
+
+<hline>
+<tline " ~mean loss~loss std~n~auc~oneoff auc">
+<hline>
+
+<tline yes~0.280~0.037~4~0.553~0.522>
+<tline no~0.277~0.0335~3~0.571~0.528>
+
+<hline>
+</table>
+
diff --git a/data/03graphae/04identityproblems b/data/03graphae/04identityproblems
index 4a6a78b8e8e081e9ad3d8fe1ca9430ad5a485525..15c121c404a6078bb3aaa238d0d0a2fdf0e22c22 100644
--- a/data/03graphae/04identityproblems
+++ b/data/03graphae/04identityproblems
@@ -1,4 +1,4 @@
-<subsection Building Identities out of Graphs>
+<subsection title="Building Identities out of Graphs" label="identities">
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 (ENTER CHAPTER), the Graph Update Step is given by
##f(x_i*s_j+x_i*A_k**i*n_j**k)##
diff --git a/data/03graphae/04scaling b/data/03graphae/04scaling
index ce17e2d9f5244e0e466399aa21a6162603e6c590..dd95432d49abe8d48e7c41fda6945091675054d5 100644
--- a/data/03graphae/04scaling
+++ b/data/03graphae/04scaling
@@ -1,4 +1,4 @@
-<subsection Scaling>
+<subsection title="Scaling" label="scaling">
NOT AT THE RIGTH POSITION
@@ -7,13 +7,13 @@ Given working small models, you migth think, that creating a bigger model is fai
<i f="trivscale" wmode="True">(mmt/trivscale)scaling plot for non norms</i>
Why is that? C addition (see Chapter (ENTER CHAPTER)): The Network focusses on each Part of the Jet the same, but in the Particles with higher #p_T# there is more Information, that gets watered down, by adding less accurate Information to it.
-<subsubsection Combination for Scaling>
+<subsubsection title="Combination for Scaling" label="scalcomb">
The easiest way to understand this, is by splitting up the Network into small Networks. Here we use #n# #4# Particle Networks, instead of one #4*n# Particle Network. Given the sum of those Networks, Scaling is still a Problem
<i f="none" wmode="True">no norm split scaling, but no c add</i>
but we now can use C addition to fix the focus of the Network: We can approximate that the mean of the signal is proportional to the width of the background distributions, and thus divide each partial Network by its loss to the power #-3#, which results in a Network that actually improves by adding more particles
<i f="splitscale" wmode="True">(mmt/splitscale, not perfect since shape and triv compare already) no norm split scaling with c add</i>
-<subsubsection Redefining losses>
+<subsubsection title="Redefining losses" label="scalloss">
One thing to remember here, is that these Networks are still just a combination of trivial Models. So the obvious question is how to apply this to bigger Networks? The first Idea migth be to redefine the loss, in a way, that the loss of any particle get multiplied by the factors used in the splitted Networks. The Problem here is that this chances the focus of the Autoencoder again: Since then it can make more errors in the later particles, it will, making the later Particles less useful for classification, but also making the first Particles more useless, since the focus is now lying on them, making their reconstruction better, up to a point at which their reconstruction is to good to find any difference between Background and Signal.
Another Idea migth be to just apply this loss weigthing in the Evaluation Phase, and not in Training. This definitely helps, but in my tries does not seem to be enough, since the same effect as before works now in the opposite way: Particles with lower #p_T# have a higher inaccuracy, that translates to a higher loss for them, and the Autoencoder focussing more on them, making the lower Particles and the higher Particles less useful.
So it seems that this is an optimization Problem: There still seems to be an optimal loss weigthing that gives optimal contribution to each Part, but finding this combination is not trivial. We tried multiple functions including a loss that uses the Index of the Particle or a loss that is weigthed directly by the transverse Momentum, but we have not found anything that mostly works, and weigthed Networks always seem to result in worse looking reconstruction
diff --git a/data/03graphae/05losses b/data/03graphae/05losses
index ce4b837659535aa94c4e85b70b606ba4b7c28b41..06202b91b27cefbdde005e6132f3bdd106782461 100644
--- a/data/03graphae/05losses
+++ b/data/03graphae/05losses
@@ -1,14 +1,14 @@
-<subsection Losses>
+<subsection title="Losses" label="losses">
Choosing what the Network focusses on, is done through two things: The Inital Normalisation, and the Loss Function. While the Size of the Initial Variables is relatively straigth forward<note The bigger Each Value, the bigger its contribution to the loss>, choosing different loss functions makes less predictable chances, so I will discuss some different Methods here
-<subsubsection L2>
+<subsubsection title="L2" label="l2">
Beeing probably the most obvious choice, setting the loss function to be the quadratic difference between Input and Output still has some benefits
##Eq(loss,(inn-out)**2)##
Not only is this easy to implement and fast to compute, but it also punishes bigger difference more than multiple small differences (guetting twice half the loss is prefered over once guetting the full loss), which generally prefered over the alternative (see subsubsection (ENTER SUBSUBSECTION)), but also results in Autoencoders that learn mean Values: In the Task of choosing #a# or #-a#, a l2 normalised Network, that does not know the rigth choise, whill choose #0#, since guessing the wrong result, is punished more, than not choosing<note You migth ask why this is a problem, since a Network that does not know anything probably should not choose one of the results, but there is a similar effect for non perfect guessing Networks: Lets say the Network guesses rigth #alpha# times, then for a prediction of #LessThan(b,a)#, the loss is given by #alpha*(a-b)**2+(1-alpha)*(a+b)**2#, which is minimal for #Eq(b,a*(2*alpha-1))#>. This results in the Fact, that the Output of a L2 Autoencoder always has a lower width than its Input. This is the central Problem, we want to solve, by looking at different losses
<i f="410simpledraw16" wmode="True">(410/simpledraw, maybe not the best image)L2 image, showing its Problems</i>
-<subsubsection Evalutating losses>
+<subsubsection title="Evalutating losses" label="evalloss">
To evaluate the Quality of a loss concerning this Problem, we define a new Parameter #varfrac# as follows:
##Eq(varfrac,min(std(A(x)**i)/std(x**i)))##
where #std(x)# is the standart deviation of the Input, #A(x)# is the Autoencoder transformation, #x**i# is one of the Features (so either an angle or #p_T#) and the #min(x**i)# function chooses the minimum Value over each #i#.
@@ -16,7 +16,7 @@ This Value is generally between 0 and 1. A Value of #0# would mean, that the Net
-<subsubsection Ln>
+<subsubsection title="Ln" label="ln">
The first other kind of loss you can look at, would be a Ln loss. For each #LesssThan(2,n)#, this Loss still has the same Problem of uncertain Losses, but for smaller n it does not. L1 does not prefer lower Predictions<note the loss described above would now look like #Eq(alpha*a-b+(1-alpha)*a+b,a+b*(1-2*alpha))# which is minimal (for each #LessThan(0.5,alpha)# for b beeing as big as possible (this loss assumes #LessThan(b,a)#, and anything Guessing rigth less than half of the Time, would still result in the Network learning not to guess, as we would want>, and a n lower than 1 would reverse the Effect entirely<note We tried #Eq(n,1/2)#, but this results in NANs, so we had to tweak #sqrt(abs(a-b))# into #sqrt(1+abs(a-b))-1# since the NANs are a result of the square root not beeing differentiable at 0. Using this Tweak, the same loss looks like #alpha*sqrt(1+a-b)+(1-alpha)*sqrt(1+a+b)-2# (ENTER MINIMIZATION)>. And in a sense, this works, those Networks have the same Input width as Output width, but the different loss has another effect: Since now one big loss is at most as worse as two small ones, Networks, that remember some Values exactly, while guessing the remaining ones, are common.
<i f="none" wmode="True">Remember 3 Networks</i>
You could ask yourself if this is even a problem. Since these Networks are not just setting the remaining Values to some constant, but try to guess them rigth, we has the same situation as in the Normal Case: On the known Data, this guessing works, and on abnormal Data this works less good. But not only do less accurate guesses have way less Decision Power<note You can model this, as a fixed distance between two gaussian Peaks with variying width, a plot of the relation between the width and the AUC is shown in (ENTER LINK)>, this also ignores the remaining Values completely: Since copying if easily done even if the Data is abnormal, there is no information gained here. And even if it were, this difference would be tiny compared to the loss of the guessed Values, so the whole loss is dominated by only some inaccurate Values, and this is a Problem. This does not mean, that LN losses are useless, since there are Networks without this Problem, but retraining each Network, until it does not do this, makes this Loss less desirable
@@ -24,7 +24,7 @@ You could ask yourself if this is even a problem. Since these Networks are not j
-<subsubsection Image like losses>
+<subsubsection title="Image like losses" label="imageloss">
One thing to consider, is that Image like Networks usually do not have these Problems. The main difference in losses, is the fact, that Imagelike Networks do not compare all Values to each other, but only one Value, if the other Values match. A pixel is restored with zero loss, if its angles are about correct, and the Momentum is correct. If the angles are a bit to much off, the loss is derived by comparing the Input to the reconstructed zero, and the newly constructed Momentum to the zero of a pixel that was initialy zero. This means, that each difference in angles, is punished the same, and the only way, for the Network to improve, is to guess those rigth and if we could implement this here, this would solve the Problem of guessing to low, at least in angles<note The Problem stays the same for #p_T#, since guessing wrongly zero is punished way less, than guessing wrongly #a#. So for the same setup as before, the new loss looks exactly the same as in the Ln case #alpha*(a-b)**n+(1-alpha)*(a+b)**n#>. So how to implement this: The first Idea, would be to set the #p_T# reconstruction to zero, if the angles are more than a certain difference apart. This works fine, but offers no control and for implementation purposes, a symmetric loss function would be nice. We choose the following extension
##f(d)*(p_T**a-p_T**b)+(1-f(d))*c(x)##
where #f(d)# is some function of the angular difference, so for example a step function (#Eq(f(d),1)# for #LessThan(d,d_0)# and #Eq(f(d),0)# else) and #c(x)# is some alternative loss, for example #Eq(c(x),abs(p_T**a+p_T**b))#. This offers a lot more flexibility and is symmetric as long as #c(x)# is symmetric.
diff --git a/data/03graphae/06caddition b/data/03graphae/06caddition
index d0917a346a5bbacf9c334a7a60a2f967b76bae21..634e3948b06f70350847edb809fb38e77e90665b 100644
--- a/data/03graphae/06caddition
+++ b/data/03graphae/06caddition
@@ -1,4 +1,4 @@
-<subsection C addition>
+<subsection title="C addition" label="caddition">
PROBABLY NOT AT THE CORRECT POSITION
diff --git a/data/03graphae/08whygae b/data/03graphae/08whygae
index 52d4e132be4a1b88fa1bd9175360d77ccbf187bd..d938919b7fecbb2bb2f9ffb42c5f6fc8b2b1f52e 100644
--- a/data/03graphae/08whygae
+++ b/data/03graphae/08whygae
@@ -1,4 +1,4 @@
-<subsection Why use Graph Autoencoder>
+<subsection title="Why use Graph Autoencoder" label="why gae">
NOT AT THE RIGTH POSITION
diff --git a/data/04aucsinv/01intro b/data/04aucsinv/01intro
index eac29e543a9cb6a80bdbb197fbb9b60fcefa92f9..7f24d6b92f7426deb9c3eb7bc8f3da112dc90f20 100644
--- a/data/04aucsinv/01intro
+++ b/data/04aucsinv/01intro
@@ -1,2 +1,2 @@
-<section Auc Scores and Invertibility>
+<section title="Auc Scores and Invertibility" label="secinv">
diff --git a/data/04aucsinv/02simplicity b/data/04aucsinv/02simplicity
index 4e6f5baf3b8ccff803822257c87eed5a197c448c..af11458303e01d5b32902014443c0773789ae79e 100644
--- a/data/04aucsinv/02simplicity
+++ b/data/04aucsinv/02simplicity
@@ -1,4 +1,4 @@
-<subsection alternative models>
+<subsection title="Simplicity" label="simplicity">
NOT THE BEST POS WS
diff --git a/data/04aucsinv/03inv b/data/04aucsinv/03inv
index f3a79a3c0a41ed06d7a00e5f56cff20e54325677..4edb09d8b6dabb128d8f1345a037d59411839120 100644
--- a/data/04aucsinv/03inv
+++ b/data/04aucsinv/03inv
@@ -1,4 +1,4 @@
-<subsection Problems in Invertibility>
+<subsection title="Problems in Invertibility" label="inv">
Given a Classifier that if trained on qcd, finds top jets different, you could (and should) ask yourself if this is just a feature of your data, as shown in Chapter (ENTER CHAPTER) or if this is more general. The easiest way to test this, is just to swithc the meaning of signal and background: Can an Autoencoder trained on top jets classify qcd jets as different. To keep those plots easily readable, I keep qcd as background, which results in an AUC score of 0 beeing optimal for those switched Networks, but when you look at a simple model, it is not
<i f="none" wmode="True">invertibility of an old model</i>
diff --git a/data/04aucsinv/04norm b/data/04aucsinv/04norm
index 758f3499846bc1f9d4a244b0007906763624c1c1..b1f99a6237ab321e4fdd7152d2267fd6ba374bce 100644
--- a/data/04aucsinv/04norm
+++ b/data/04aucsinv/04norm
@@ -1,12 +1,12 @@
-<subsection solving Invertibility through Normation>
+<subsection title="solving Invertibility through Normation" label="normation">
-<subsubsection The Meaning of Complexity>
+<subsubsection title="The Meaning of Complexity" label="complexity">
Since there is a Model, that can reach a similar quality as an good Autoencoder, by just comparing the Angular Data to zero, it stands to reason, that these Autoencoder work in a similar way, which of course would not make them Invertible<note a model comparing just angles to zero, would be the same, if trained on qcd or on top jets>.
We can test this Hypothesis, by just removing any kind of Data difference, that allows for this trivial comparison, which, as shown in Chapter (ENTER CHAPTER) is the width of the Input Distribution, or in other words: What we need is some kind of Normation.
This method has one obvious drawback: Not only do we actively remove Information, which will hurt the Performance, but we remove the Information, that generates most of the quality. This means, even if the resulting Classifier is invertible, it will probably be way worse than the trivial one.
-<subsubsection How to normalise an Autoencoder>
+<subsubsection title="How to normalise an Autoencoder" label="normprobs">
One thing, I did not realise after triying to normalise my Input Datapoints, is that simply demanding that the mean is zero and the standart deviation is one, 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 Chapter (ENTER CHAPTER ONEOFF). The Problem is, that by demanding a value to be fixed, we remove one information from 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 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<note ignoring flag for now, three Values is always enough to encode 4 flag Values, since the first three flag Values are always one (the jet with the lowest number of particles has 3???? particles in my trainingsset)>. In reality things are not so easy. There is no garantee that this minima is found, and the Graph Structure does not neccesarily allow for this kind of transformation(see Chapter (ENTER CHAPTER)), but training this kind of Network, does not result in the Model gaining any Classification Power. This can be seen in the corresponding Feature Map
<i f="none" wmode="True">ABE for standart norm networks, showing nothing useful</i>
This migth seem now, as if there is a trivial solution: just reduce the compression size accordingly, but this has three problems
@@ -36,7 +36,7 @@ Here the definitions of #y# and #n# assert Translation and Scale Invariance resp
<i f="none" wmode="True">ABE for better norm</i>
-<subsubsection Using this normation>
+<subsubsection title="Using this normation" label="usenorm">
Using this kind of Normation, at least 4 node Networks are invertible. And not only this, but also each Feature is Invertible.
<i f="none" f2="none" wmode="True">invertible 4node network auc maps</i>
But the Quality suffers
@@ -58,7 +58,7 @@ training for 1000 epochs, and then until the loss increases for 250 epochs, resu
-<subsubsection Improving the Normation even further>
+<subsubsection title="Improving the Normation even further" label="normplus">
After seeing what an effect some kind of Normation can have, we are not completely satisfied anymore with the normated Feature Maps:
<i f="none" wmode="True">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 immediatly becomes clear why
diff --git a/data/04aucsinv/06aucs b/data/04aucsinv/06aucs
index fae9e975d0c306370ebd94dd41afc0d1a7c9cd15..80a92447c5d2c45090e45900037bd59d287a3847 100644
--- a/data/04aucsinv/06aucs
+++ b/data/04aucsinv/06aucs
@@ -1,4 +1,4 @@
-<subsection Auc Scores for Toptagging>
+<subsection title="Auc Scores for Toptagging" label="aucfortt">
HAVE I ALREADE WRITTEN THIS?
diff --git a/data/05otherdata/01intro b/data/05otherdata/01intro
index 9bfde9a464f7ca3963b407df01c12390cef26046..0c31e18852253519e6d98a14b38a6323b297f31e 100644
--- a/data/05otherdata/01intro
+++ b/data/05otherdata/01intro
@@ -1,2 +1,2 @@
-<section Data>
+<section title="Data" label="secdata">
diff --git a/data/05otherdata/02ldm b/data/05otherdata/02ldm
index 010ef1b0eea9d13ad3ddb3212836028b767fe7fc..65acdd4f08eeeb10b9e5d909178186ef1d5c66c5 100644
--- a/data/05otherdata/02ldm
+++ b/data/05otherdata/02ldm
@@ -1,4 +1,4 @@
-<subsection Ligth Dark Matter>
+<subsection title="Ligth Dark Matter" label="ldm">
this chapter is missing, since it is still beeing worked on
diff --git a/data/05otherdata/03quarkgluon b/data/05otherdata/03quarkgluon
index ed5b53559c929b731bffddb52f90ab3f5e42038d..58922a6da719447050e5c5eea47943c451c6e4c0 100644
--- a/data/05otherdata/03quarkgluon
+++ b/data/05otherdata/03quarkgluon
@@ -1,3 +1,3 @@
-<subsection Quark v Gluon>
+<subsection title="Quark v Gluon" label="qg">
this chapter is missing, since it is still beeing worked on
diff --git a/data/06othernets/01title b/data/06othernets/01title
index d9489b21d7d5b4b468743447fe3847284ddf709b..89c00c7062d1680a1e5f7f606ef65af58ee77107 100644
--- a/data/06othernets/01title
+++ b/data/06othernets/01title
@@ -1,2 +1,2 @@
-<section Other Approaches>
+<section title="Other Approaches" label="secother">
diff --git a/data/06othernets/03oneoff b/data/06othernets/03oneoff
index 90d1f4ff8a5bcb6caf1e1ff3b8d2e1efcc8b6060..29a9835647fcb21167b555cc5420c783a4b19f2c 100644
--- a/data/06othernets/03oneoff
+++ b/data/06othernets/03oneoff
@@ -1,4 +1,4 @@
-<subsection Oneoff Networks>
+<subsection title="Oneoff Networks" label="oneoff">
When you consider the following Feature Map of a well normated Network:
<i f="aucmapb" wmode=True>(534)AUC Feature Map for an on Top trained Autoencoder, using a good normation</i>
@@ -6,7 +6,7 @@ You migth notice, that most of the decision power is in the first feature, but t
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 compressed 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 has no physical Meaning. More likely is the following: The Network saves one Value in the Compression Space, and is able to reconstruct an one from all the other Parameters. This makes sence, since we got this Auc Distribution by chancing the Normation 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 thus by chancing the Inputs, the constant is probably 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</note>, and on Paper it seams like a great Idea: As shown before (see Chapter (ENTER CHAPTER)), complexity is to a big part just width. You may be able to solve this by Normation, 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 normation 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. This results in very high correlations in the Outputs, but seems to simplify the convergence</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 can result in the Network beeing able to learn more complicated functions.
-<subsubsection oneoff math>
+<subsubsection title="oneoff math" label="oomath">
Before we talk about the results, lets talk about the math 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 (ENTER LINK), and it is true, that by considering how to combine two Background, the Math is the 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 (ENTER LINK). This migth suggest, that the resulting combination of an OneOff Network is the combination with the highest possible AUC. Sadly this is simply not true, so where did this go wrong: The Problem are the assumptions made in C addition: 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 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 distance could be some sort of signal Data, that you want to catch. 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 focus width a width #LessThan(s_2,s_1)#: Which Peak is more probable to seperate any signal from the Background? Since the second Peak is less wide, it is less probable for a Signal Peak to overlap it, than to overlap the bigger Peak, and thus the better optimized Peak, 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 chances 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 is useful</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>
@@ -20,7 +20,7 @@ THE ABOVE MATH STILL NEEDS SOME DOUBLECHECKING
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 readjust 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 they are not always the same. In the following mostly readjusted means are used.
-<subsubsection oneoff quality>
+<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<note>#10**(-12)#</note>, 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 in combinations of 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. A notable example migth be the preprocessing of #p_T#. As descibed in Chapter (ENTER CHAPTER), 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#. <note>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>. Since I usually use Relu activations (This is another thing where those Networks can become trivial, think of a sigmoid and a Network just learning infinity), 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 (ENTER CHAPTER), this is no longer needed, and just complicates the Training</note>.
@@ -29,7 +29,7 @@ Now for Numbers: A simple oneoff Network reaches usually an Auc of about #0.6#,
Sadly, this observation is not really useful, since stopping the Training at the optimal Epoch would not be unsupervised, but it is 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 (ENTER CHAPTER)</note>.
Another Problem is again invertibility: It is possible to create an invertible oneoff Network, but it is not trivially given. They work best, when you use a certain minimal complexity. To do this, a Graph Network seems to be less useful, than just a simple dense Network.
-<subsubsection oneoff outside of physics>
+<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 (ENTER REFERENCE)</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 my results in the Appendix (ENTER APPENDIX), 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 (ENTER CHAPTER)) 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>
diff --git a/data/06othernets/05otheralgo b/data/06othernets/05otheralgo
index 1c90af01871f48d2a5983c6fe134eb18cf8b4b51..c9a6719f5c0e1b3cf71b921b02e65a205783b80e 100644
--- a/data/06othernets/05otheralgo
+++ b/data/06othernets/05otheralgo
@@ -1,16 +1,16 @@
-<subsection Other Algorithms>
+<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.
-<subsubsection Support Vector Machines>
+<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, as it keeps the SVM from overfitting, but it also only allows it to learn certain distributions, so distributions like
<i f="img_circle" wmode="True">(c4/16) A simple example of a shape, that an SVM cannot differentiate</i>
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 ???).
-<subsubsection k neirest neighbours>
+<subsubsection title="k neirest neighbours" label="whatsnvm">
Another usually quite useful algoritm is also an extension of the 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(ENTER IMAGE), with an quite good AUC of ????<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.
-<subsubsection Isolation Forests>
+<subsubsection title="Isolation Forests" label="whatsiforest">
An Isolation Forest<note (ENTER REFERENCE)> 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 classifiyng 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 ???.
diff --git a/data/07oneoffplus/00intro b/data/07oneoffplus/00intro
index 3c63c41e82b5ea0569245600ad49e0358601baa1..0e7e7bf1f2dbffea769b27f8c75ef8f8e19e7fae 100644
--- a/data/07oneoffplus/00intro
+++ b/data/07oneoffplus/00intro
@@ -1 +1 @@
-<section Mixed Approaches>
+<section title="Mixed Approaches" label="secmixed">
diff --git a/data/07oneoffplus/01idea b/data/07oneoffplus/01idea
index 03a5776758a8e5bffb50ef6bb9aa296afa6718b2..20f9efbadbc997713421c57b8fafb278e0c1af56 100644
--- a/data/07oneoffplus/01idea
+++ b/data/07oneoffplus/01idea
@@ -1,4 +1,4 @@
-<subsection Compressed One Class Learning>
+<subsection title="Compressed One Class Learning" label="mixedidea">
The main Problem of Autoencodern is the fact that its loss function is not neccesarily the best seperator<note see Chapter (ENTER CHAPTER) and CHAPTER (ENTER CHAPTER)>, while the Problem of other Methods is the fact that they are not well equiped to handle symmetries of certain kind or trivial Information<note In the case of oneoffs>, so maybe combining both Methods is not a bad idea: You train an Autoencoder just to convert the Input Space into the compressed Space, to run other algorithms on this compressed Space. This means that the seperator is now usually quite good, and that the Autoencoder can filter out symmetries and trivial Inputs. This Idea is not neccesarily new<note see (ENTER REFERENCE)> and becomes harder to train, since both Networks have to be trained well, but this usually works quite well.
One Question you could ask yourself, if you want to train your Autoencoder on the background data or on both the signal and the background. Here mostly training on background will be used, 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.
diff --git a/data/07oneoffplus/02diffalgo b/data/07oneoffplus/02diffalgo
index d49c33f3a9d8db706ebe3c132b06566111c33fc9..8a48baef958ffea54f0ae947fa539d49be7d6e50 100644
--- a/data/07oneoffplus/02diffalgo
+++ b/data/07oneoffplus/02diffalgo
@@ -1,20 +1,20 @@
-<subsection Different Algorithms>
+<subsection table="Different Algorithms" label="mixedalt">
Here I test different One Class Algorithms on one Autoencoder that trains on the first 4 Particles of top jets and tries to find qcd jets as a signal. For comparison, simply looking at the loss of this Network, you get an AUC score of about #0.377# and the best AUC score I have seen for an Autoencoder is about #0.25#. I choose this one, since its reconstruction seems to be quite accurate, meaning that most information about the Jet is still contained in the Feature Space
<i f="none" wmode="True">677 simpledraw</i>
-<subsubsection SVM>
+<subsubsection title="SVM" label="mixedSVM">
A SVM works better on the compressed Space, now reaching an AUC of #0.434#, but even though this is definitely an improvement, it is still worse than just using the Autoencoder
-<subsubsection Isolation Forest>
+<subsubsection title="Isolation Forest" label="mixediforest">
Also the Isolation Forest improves, now reaching an AUC of #0.377#, so it is at least as good as simply using the Autoencoder.
-<subsubsection k neirest neighbour>
+<subsubsection title="k neirest neighbour" label="mixedknn">
K neirest neighbour is the first algorithm that improves over simply using the Autoencoder loss, reaching an AUC of #0.307#, even though it should be noted, that this is for #Eq(k,1)# and gets worse for higher #k#. Also this is still worse than our best Autoencoder.
-<subsubsection oneoff>
+<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 my autoencoder, and by combining multiple ones, you can reach an AUC of about #0.2# beeing quite good.
diff --git a/data/07oneoffplus/08whyaesuck b/data/07oneoffplus/08whyaesuck
index e00ac1e4701dcba03c8d2314d647b197eb04567c..48e9f2cb5054f6bf0c0f877cc3004f17b3ac1ea2 100644
--- a/data/07oneoffplus/08whyaesuck
+++ b/data/07oneoffplus/08whyaesuck
@@ -1,8 +1,8 @@
-<subsection Why Autoencoder might not be so great>
+<subsection title="Why Autoencoder might not be so great" label="whynotgae">
NOT AT THE RIGTH POSITION
-<subsubsection Reproduding vs Classifing Quality>
+<subsubsection title="Reproduding vs Classifing Quality" label="nogaequal">
When I started working on Anomaly Detection, I thougth that Autoencoder 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. This you can do 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, so after having a bit more experience encoding Data to classify it, and especcially when we now have another Method of seperating Data, I want to start this Chapter about why Autoencoder migth not be the best Idea, evaluating this hypotheosis.
To do this, lets look first at the loss of the Autoencoder: Since it is basically just the difference between Input and Output, it is a Measurement about how good the Autoencoder reproduces whatever we put into it<note This is a bit of a simplification, since the Normation 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. Also, as argued for in the Chapter about Decicision Quality (ENTER CHAPTER LINK), we will use the AUC score to evalutate the Classifier Quality. So basically we want to see a strong falling correlation between the loss of my Network and the AUC score<note The lower the loss, the higher the AUC>, do we see this? We show here the loss in training against the Decicion 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 cancelled by the Network not always reaching 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 W 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">A simple old ABE nonorm focus on angular</i>
@@ -26,13 +26,13 @@ As a final note in this subchapter: Please consider that oneoff networks should
(one could also note, that the same curve looks factually the same for a nongraph one, but less beautiful,but one could also not do this, since this is a point for graph oneoffs and I am not using graph oneoffs)
-<subsubsection General Problems>
+<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, I want to focus here on two:
Firstly, it is not trivial to find the best compression size. The lower this size, the more Feature migth be compared trivially, and the higher it is, the more Features migth be reconstructed trivially<note here the graph structure actually helps, since permutation invariance makes it nontrivial to get an true identity>. I choose my compressionsize by observing that a 4 node network is only invertible for a compression size over 9<note 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 I just leave the Fraction constant. You migth ask why I dont just test this again for a higer node count, but this would not only be quite timeintensive, but also a little chance does not really matter.
and also there is this image (circle reconstruction problem compared to oneoffs)
-<subsubsection Autoencoder and losses>
+<subsubsection title="Autoencoder and losses" label="nogaeloss">
talk about the mean guessing nets
diff --git a/data/08anhang/01title b/data/08anhang/01title
index d7be0f9c371dfde19ad4c40fc661199441138c1e..a509eb6dd1f8fb3280d44eeba38ee5be01f469be 100644
--- a/data/08anhang/01title
+++ b/data/08anhang/01title
@@ -1,2 +1,2 @@
-<section Appendix>
+<section title="Appendix" label="appendix">
diff --git a/out/label.json b/out/label.json
new file mode 100644
index 0000000000000000000000000000000000000000..5c4ac88cef2e252a2ec20d6ea1eb3a546c6d04ce
--- /dev/null
+++ b/out/label.json
@@ -0,0 +1,1063 @@
+[
+ {
+ "typ": "section",
+ "title": "Introduction and Literature",
+ "label": "secintro",
+ "file": "..\\..\\write\\/data\\01intro\\00title"
+ },
+ {
+ "typ": "subsection",
+ "title": "Introduction",
+ "label": "intro",
+ "file": "..\\..\\write\\/data\\01intro\\01intro"
+ },
+ {
+ "typ": "subsection",
+ "title": "New Physics",
+ "label": "physics",
+ "file": "..\\..\\write\\/data\\01intro\\02physics"
+ },
+ {
+ "typ": "subsection",
+ "title": "Autoencoder",
+ "label": "ae",
+ "file": "..\\..\\write\\/data\\01intro\\03ae"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/linear"
+ ],
+ "label": "linear",
+ "caption": "(mmt/q/02/linear) linear 2d sample data for ae"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "sample combinations of two image Inputs (something like cats and dogs)"
+ },
+ {
+ "typ": "subsection",
+ "title": "Anomaly Detection",
+ "label": "anoma",
+ "file": "..\\..\\write\\/data\\01intro\\04anomalydetection"
+ },
+ {
+ "typ": "subsection",
+ "title": "Graphs",
+ "label": "graphs",
+ "file": "..\\..\\write\\/data\\01intro\\05graphs"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/dia3"
+ ],
+ "label": "dia3",
+ "caption": "(mmt/dia3) a simple node edge explanaiton graph"
+ },
+ {
+ "typ": "subsection",
+ "title": "Graph Autoencoder",
+ "label": "gae",
+ "file": "..\\..\\write\\/data\\01intro\\06graphae"
+ },
+ {
+ "typ": "section",
+ "title": "Definitions",
+ "label": "secdef",
+ "file": "..\\..\\write\\/data\\02tech\\00intro"
+ },
+ {
+ "typ": "subsection",
+ "title": "Binary Classification",
+ "label": "binclass",
+ "file": "..\\..\\write\\/data\\02tech\\01binclass"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "ROC curve",
+ "label": "classroc",
+ "file": "..\\..\\write\\/data\\02tech\\01binclass"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Area Under the Curve",
+ "label": "classauc",
+ "file": "..\\..\\write\\/data\\02tech\\01binclass"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "other Measures",
+ "label": "classother",
+ "file": "..\\..\\write\\/data\\02tech\\01binclass"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/defttetc"
+ ],
+ "label": "defttetc",
+ "caption": "(redoo yourself) Definitions of the 4 Fractions used to evaluate binary classification"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/add3"
+ ],
+ "label": "add3",
+ "caption": "(mmt/add3, prob not perfect)some recqual with decision parameter implemented"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "a roc curve with the classical Plotting"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "The other way of plotting a roc curve"
+ },
+ {
+ "typ": "subsection",
+ "title": "Problems in Evaluating a Model",
+ "label": "eval prob",
+ "file": "..\\..\\write\\/data\\02tech\\02evmod"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "AUC scores",
+ "label": "evalauc",
+ "file": "..\\..\\write\\/data\\02tech\\02evmod"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Losses",
+ "label": "evalloss",
+ "file": "..\\..\\write\\/data\\02tech\\02evmod"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Images",
+ "label": "evalimg",
+ "file": "..\\..\\write\\/data\\02tech\\02evmod"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Oneoff width",
+ "label": "evaloow",
+ "file": "..\\..\\write\\/data\\02tech\\02evmod"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "some image, with nothing learned in pt"
+ },
+ {
+ "typ": "subsection",
+ "title": "Datapreperation",
+ "label": "data",
+ "file": "..\\..\\write\\/data\\02tech\\04dataprep"
+ },
+ {
+ "typ": "subsection",
+ "title": "Explaining Graphics",
+ "label": "graphics",
+ "file": "..\\..\\write\\/data\\02tech\\06explain"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Output Images",
+ "label": "imgout",
+ "file": "..\\..\\write\\/data\\02tech\\06explain"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "AUC Feature Maps",
+ "label": "imgmaps",
+ "file": "..\\..\\write\\/data\\02tech\\06explain"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Network Setups",
+ "label": "imgsetups",
+ "file": "..\\..\\write\\/data\\02tech\\06explain"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "a sample output input image"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "sample lpt image"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "a sample Featuremap"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "a sample Network Image, see below or probably just 900"
+ },
+ {
+ "typ": "subsection",
+ "title": "Graph Neuronal Networks",
+ "label": "gnn",
+ "file": "..\\..\\write\\/data\\02tech\\09graphnet"
+ },
+ {
+ "typ": "section",
+ "title": "Making Graph Autoencoder work",
+ "label": "secgae",
+ "file": "..\\..\\write\\/data\\03graphae\\00title"
+ },
+ {
+ "typ": "subsection",
+ "title": "Failed Approaches",
+ "label": "failed",
+ "file": "..\\..\\write\\/data\\03graphae\\01failed"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "trivial models",
+ "label": "failedtrivial",
+ "file": "..\\..\\write\\/data\\03graphae\\01failed"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "minimal models",
+ "label": "failedminimal",
+ "file": "..\\..\\write\\/data\\03graphae\\01failed"
+ },
+ {
+ "typ": "subsection",
+ "title": "An explicit look at the first working graph autoencoder",
+ "label": "firstworking",
+ "file": "..\\..\\write\\/data\\03graphae\\02simpleae"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "topK",
+ "label": "quicktopK",
+ "file": "..\\..\\write\\/data\\03graphae\\02simpleae"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Compression",
+ "label": "quickcompression",
+ "file": "..\\..\\write\\/data\\03graphae\\02simpleae"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Decompression",
+ "label": "quickdecompression",
+ "file": "..\\..\\write\\/data\\03graphae\\02simpleae"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Sorting",
+ "label": "quicksort",
+ "file": "..\\..\\write\\/data\\03graphae\\02simpleae"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Training Setup",
+ "label": "quicktrain",
+ "file": "..\\..\\write\\/data\\03graphae\\02simpleae"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Results",
+ "label": "quickres1",
+ "file": "..\\..\\write\\/data\\03graphae\\02simpleae"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "modeldraw of b1/00 (achte auf dense und das batchnorm vor compare ist)"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "compression pictograms"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "decompression pictograms"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "training for b1/00"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "b1/00/simpledraw ml, auf cherrypick hinweise"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "roc curve b1/00...nett machen"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "b1/00/partialauc raspl"
+ },
+ {
+ "typ": "subsection",
+ "title": "Improving Autoencoder",
+ "label": "secondworking",
+ "file": "..\\..\\write\\/data\\03graphae\\03goodae"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Training Setup",
+ "label": "quick2setup",
+ "file": "..\\..\\write\\/data\\03graphae\\03goodae"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Results",
+ "label": "quick2results",
+ "file": "..\\..\\write\\/data\\03graphae\\03goodae"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "network plot for c1/200"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "history for c1/200"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "roc for c1/200"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "simpledraw for c1/200 7638, maybe also the before one?"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "AUC feature map for c1/200"
+ },
+ {
+ "typ": "subsection",
+ "title": "Making Autoencoder good",
+ "label": "thirdworking",
+ "file": "..\\..\\write\\/data\\03graphae\\03optimalae"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "better encoding",
+ "label": "encoding",
+ "file": "..\\..\\write\\/data\\03graphae\\03optimalae"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "better decoding",
+ "label": "decoding",
+ "file": "..\\..\\write\\/data\\03graphae\\03optimalae"
+ },
+ {
+ "typ": "table",
+ "caption": "Quality differences for different encoder with a learnable handling of the Feature vectors",
+ "label": "encode1",
+ "file": "..\\..\\write\\/data\\03graphae\\03optimalae"
+ },
+ {
+ "typ": "table",
+ "caption": "Quality differences for different encoder with a fixed function",
+ "label": "encode2",
+ "file": "..\\..\\write\\/data\\03graphae\\03optimalae"
+ },
+ {
+ "typ": "table",
+ "caption": "Quality differences for different graph like decoder",
+ "label": "decode1",
+ "file": "..\\..\\write\\/data\\03graphae\\03optimalae"
+ },
+ {
+ "typ": "table",
+ "caption": "Quality differences for different param like decoder",
+ "label": "decode2",
+ "file": "..\\..\\write\\/data\\03graphae\\03optimalae"
+ },
+ {
+ "typ": "table",
+ "caption": "Quality difference for either running learnable sub graph updates or not",
+ "label": "decode3",
+ "file": "..\\..\\write\\/data\\03graphae\\03optimalae"
+ },
+ {
+ "typ": "subsection",
+ "title": "Building Identities out of Graphs",
+ "label": "identities",
+ "file": "..\\..\\write\\/data\\03graphae\\04identityproblems"
+ },
+ {
+ "typ": "subsection",
+ "title": "Scaling",
+ "label": "scaling",
+ "file": "..\\..\\write\\/data\\03graphae\\04scaling"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Combination for Scaling",
+ "label": "scalcomb",
+ "file": "..\\..\\write\\/data\\03graphae\\04scaling"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Redefining losses",
+ "label": "scalloss",
+ "file": "..\\..\\write\\/data\\03graphae\\04scaling"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/trivscale"
+ ],
+ "label": "trivscale",
+ "caption": "(mmt/trivscale)scaling plot for non norms"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "no norm split scaling, but no c add"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/splitscale"
+ ],
+ "label": "splitscale",
+ "caption": "(mmt/splitscale, not perfect since shape and triv compare already) no norm split scaling with c add"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "reconstruction comparison for weigthed losses"
+ },
+ {
+ "typ": "subsection",
+ "title": "Losses",
+ "label": "losses",
+ "file": "..\\..\\write\\/data\\03graphae\\05losses"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "L2",
+ "label": "l2",
+ "file": "..\\..\\write\\/data\\03graphae\\05losses"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Evalutating losses",
+ "label": "evalloss",
+ "file": "..\\..\\write\\/data\\03graphae\\05losses"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Ln",
+ "label": "ln",
+ "file": "..\\..\\write\\/data\\03graphae\\05losses"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Image like losses",
+ "label": "imageloss",
+ "file": "..\\..\\write\\/data\\03graphae\\05losses"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/410simpledraw16"
+ ],
+ "label": "410simpledraw16",
+ "caption": "(410/simpledraw, maybe not the best image)L2 image, showing its Problems"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "Remember 3 Networks"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "L1 image, working"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "Image like loss working well"
+ },
+ {
+ "typ": "subsection",
+ "title": "C addition",
+ "label": "caddition",
+ "file": "..\\..\\write\\/data\\03graphae\\06caddition"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/dist1"
+ ],
+ "label": "dist1",
+ "caption": "(mmt/dist1)double Gaussian Peak quality example"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "c addition example"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "c addition on jets for 4 parts once with power -3 and one with power 0"
+ },
+ {
+ "typ": "subsection",
+ "title": "Why use Graph Autoencoder",
+ "label": "why gae",
+ "file": "..\\..\\write\\/data\\03graphae\\08whygae"
+ },
+ {
+ "typ": "section",
+ "title": "Auc Scores and Invertibility",
+ "label": "secinv",
+ "file": "..\\..\\write\\/data\\04aucsinv\\01intro"
+ },
+ {
+ "typ": "subsection",
+ "title": "Simplicity",
+ "label": "simplicity",
+ "file": "..\\..\\write\\/data\\04aucsinv\\02simplicity"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "AUC map for non normated nets"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "Input size vs output size for each Feature and non normated nets"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "2d hist angular of qcd vs top"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "scaling of zero comparison no c addition"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "scaling of zero comparison with c addition"
+ },
+ {
+ "typ": "subsection",
+ "title": "Problems in Invertibility",
+ "label": "inv",
+ "file": "..\\..\\write\\/data\\04aucsinv\\03inv"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "invertibility of an old model"
+ },
+ {
+ "typ": "subsection",
+ "title": "solving Invertibility through Normation",
+ "label": "normation",
+ "file": "..\\..\\write\\/data\\04aucsinv\\04norm"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "The Meaning of Complexity",
+ "label": "complexity",
+ "file": "..\\..\\write\\/data\\04aucsinv\\04norm"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "How to normalise an Autoencoder",
+ "label": "normprobs",
+ "file": "..\\..\\write\\/data\\04aucsinv\\04norm"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Using this normation",
+ "label": "usenorm",
+ "file": "..\\..\\write\\/data\\04aucsinv\\04norm"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Improving the Normation even further",
+ "label": "normplus",
+ "file": "..\\..\\write\\/data\\04aucsinv\\04norm"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "ABE for standart norm networks, showing nothing useful"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "ABE for better norm"
+ },
+ {
+ "typ": "img2",
+ "files": [
+ "../imgs/none",
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "invertible 4node network auc maps"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "double roc curve for invertibility of normated networks"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "1k Networks, loss vs auc plot"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "justification of the new compression size, auc vs quality for compression size 8 and 9"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "comparison of reproducability for earlystopping and minepoch"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "ABE for better norm (just a copy)"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "better norm pt0 dist"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "ABE good norm"
+ },
+ {
+ "typ": "subsection",
+ "title": "Auc Scores for Toptagging",
+ "label": "aucfortt",
+ "file": "..\\..\\write\\/data\\04aucsinv\\06aucs"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "Invertibility as function of compression, showing also in accuracy"
+ },
+ {
+ "typ": "section",
+ "title": "Data",
+ "label": "secdata",
+ "file": "..\\..\\write\\/data\\05otherdata\\01intro"
+ },
+ {
+ "typ": "subsection",
+ "title": "Ligth Dark Matter",
+ "label": "ldm",
+ "file": "..\\..\\write\\/data\\05otherdata\\02ldm"
+ },
+ {
+ "typ": "subsection",
+ "title": "Quark v Gluon",
+ "label": "qg",
+ "file": "..\\..\\write\\/data\\05otherdata\\03quarkgluon"
+ },
+ {
+ "typ": "section",
+ "title": "Other Approaches",
+ "label": "secother",
+ "file": "..\\..\\write\\/data\\06othernets\\01title"
+ },
+ {
+ "typ": "subsection",
+ "title": "Oneoff Networks",
+ "label": "oneoff",
+ "file": "..\\..\\write\\/data\\06othernets\\03oneoff"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "oneoff math",
+ "label": "oomath",
+ "file": "..\\..\\write\\/data\\06othernets\\03oneoff"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "oneoff quality",
+ "label": "ooquality",
+ "file": "..\\..\\write\\/data\\06othernets\\03oneoff"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "oneoff outside of physics",
+ "label": "oomnist",
+ "file": "..\\..\\write\\/data\\06othernets\\03oneoff"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/aucmapb"
+ ],
+ "label": "aucmapb",
+ "caption": "(534)AUC Feature Map for an on Top trained Autoencoder, using a good normation"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/img_doublepeak"
+ ],
+ "label": "img_doublepeak",
+ "caption": "(c4/16) Two different widths of a Background Peak, resulting in different overlapping to the signal Peak"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../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"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/examples"
+ ],
+ "label": "examples",
+ "caption": "(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"
+ },
+ {
+ "typ": "subsection",
+ "title": "Other Algorithms",
+ "label": "other",
+ "file": "..\\..\\write\\/data\\06othernets\\05otheralgo"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Support Vector Machines",
+ "label": "whatssvm",
+ "file": "..\\..\\write\\/data\\06othernets\\05otheralgo"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "k neirest neighbours",
+ "label": "whatsnvm",
+ "file": "..\\..\\write\\/data\\06othernets\\05otheralgo"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Isolation Forests",
+ "label": "whatsiforest",
+ "file": "..\\..\\write\\/data\\06othernets\\05otheralgo"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/img_circle"
+ ],
+ "label": "img_circle",
+ "caption": "(c4/16) A simple example of a shape, that an SVM cannot differentiate"
+ },
+ {
+ "typ": "section",
+ "title": "Mixed Approaches",
+ "label": "secmixed",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\00intro"
+ },
+ {
+ "typ": "subsection",
+ "title": "Compressed One Class Learning",
+ "label": "mixedidea",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\01idea"
+ },
+ {
+ "typ": "subsection",
+ "title": "",
+ "label": "mixedalt",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\02diffalgo"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "SVM",
+ "label": "mixedSVM",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\02diffalgo"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Isolation Forest",
+ "label": "mixediforest",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\02diffalgo"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "k neirest neighbour",
+ "label": "mixedknn",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\02diffalgo"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "oneoff",
+ "label": "mixedoo",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\02diffalgo"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "677 simpledraw"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "comparison of all one class algos"
+ },
+ {
+ "typ": "subsection",
+ "title": "???",
+ "label": "???",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\07questionmark"
+ },
+ {
+ "typ": "subsection",
+ "title": "Why Autoencoder might not be so great",
+ "label": "whynotgae",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\08whyaesuck"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Reproduding vs Classifing Quality",
+ "label": "nogaequal",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\08whyaesuck"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "General Problems",
+ "label": "nogaegeneral",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\08whyaesuck"
+ },
+ {
+ "typ": "subsubsection",
+ "title": "Autoencoder and losses",
+ "label": "nogaeloss",
+ "file": "..\\..\\write\\/data\\07oneoffplus\\08whyaesuck"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "A simple old ABE nonorm focus on angular"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "a \"good\" abe"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "ABE nonorm like exp on trivial decompressor"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "ABE multi Network with particle/graph like, and logged x axis"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "ABE for oneoff that nice curve"
+ },
+ {
+ "typ": "img",
+ "files": [
+ "../imgs/none"
+ ],
+ "label": "none",
+ "caption": "partial Network addition, auc(gs) for factors=choch-3 und factors=1"
+ },
+ {
+ "typ": "section",
+ "title": "Appendix",
+ "label": "appendix",
+ "file": "..\\..\\write\\/data\\08anhang\\01title"
+ }
+]
\ No newline at end of file
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index 794c5262671d333cb0c422d470240af08145c390..782eb69dea45b4b4f6fd41b04223c9bf965c9e8c 100644
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diff --git a/out/main.tex b/out/main.tex
index 63b4537002ab5e1e4b0f03b4646d84943754ddf3..b9f39eceda4de60bd6201084d8ea893b827ee579 100644
--- a/out/main.tex
+++ b/out/main.tex
@@ -63,12 +63,12 @@
%from file ..\..\write\/data\01intro\00title
\newpage
-\section{Introduction and Literature}\label{sec:Introduction and Literature}
+\section{Introduction and Literature}\label{sec:secintro}
%from file ..\..\write\/data\01intro\01intro
-\subsection{Introduction}\label{sec:Introduction}
+\subsection{Introduction}\label{sec:intro}
This chapter will be written as the last one
@@ -76,6 +76,7 @@ This chapter will be written as the last one
+
Current State of this Thesis
\begin{itemize}
@@ -142,7 +143,7 @@ Current State of Chapter 6
%from file ..\..\write\/data\01intro\02physics
-\subsection{New Physics}\label{sec:New Physics}
+\subsection{New Physics}\label{sec:physics}
This Chapter is not yet written, since I want to do it justice
@@ -150,7 +151,7 @@ This Chapter is not yet written, since I want to do it justice
%from file ..\..\write\/data\01intro\03ae
-\subsection{Autoencoder}\label{sec:Autoencoder}
+\subsection{Autoencoder}\label{sec:ae}
Autoencoder are a kind of neuronal Network, in which the function that should be learned is set to the Identity, and to make this learn more than a trivial Function, a compressed state is introduced. This compressed State is of lower Dimension than the Input Space and is generated by a learnable Function called the Encoder from the Input, and serves as Input to another learnable Function, which is called the Decoder. Together, both try to reconstruct the Input. This reconstruction is usually not perfect, since the reduced Dimension does not allow to encode each possible Value perfectly, but it can be quite good, as long as the data contains some patterns. Consider the following 2 dimensional Data:
@@ -166,6 +167,7 @@ Autoencoder are a kind of neuronal Network, in which the function that should be
As you can see, to completely encode the data you would still require 2 dimensions, but you can approximately encode them into 1 dimension quite well, by using one value as encoding and the second one, in the decoder, as a linear function of this\footnote{Since the Number of trainingsamples is finite, you could map every sample into an index, and map those indices again onto the inputs, reaching a zero loss for any Input with an compression size of 1, but not only is finding such a function quite hard for a neuronal Network, it also would not be useful at all, since on any new data, the Network would not work at all. This is why these kind of functions are a part of what is called overfitting for autoencoder}.
+
This combination of a compresser and a decompresser can be quite useful in multiple ways. Ignoring the obvious task of compressing data (see (ENTER REFERENCE) for an example of an autoencoder outperforming a classical compressor), you can give the decompressor noise to generate new versions of an already known kind of data (see (ENTER REFERENCE)), and even though nowadays GANs are used for this Task, autoencoder have still some benefits, allowing for more control over the generated Data. This works, since (in good autoencoders\footnote{This works better in a special way of training an Autoencoder, called a Variational Autoencoder.}) similarity in the compressed Space represent similarity of the Inputs, so not only by changing the compressed Versions of an Input sligthly you can reproduce similar Inputs, but by Identifieng Features in the Input Space you can chance just one attribute of an Input, and you can also merge the Features of two Inputs into one, see for these Applications (ENTER REFERENCE) and (ENTER REFERENCE)
\begin{figure}[H]
\centering
@@ -183,7 +185,7 @@ That beeing said, the Application that is focussed on in this work, is the Detec
%from file ..\..\write\/data\01intro\04anomalydetection
-\subsection{Anomaly Detection}\label{sec:Anomaly Detection}
+\subsection{Anomaly Detection}\label{sec:anoma}
Anomaly Detection, as the Task of finding abnormal Events has a plentitude of Use cases: from improving the Purity of a dataset (ENTER REFERENCE) to fraud and fault detection (see (ENTER REFERENCE) and (ENTER REFERENCE)).
@@ -193,7 +195,7 @@ To achieve this, there is a multitude of algorithms, that will be introduced and
%from file ..\..\write\/data\01intro\05graphs
-\subsection{Graphs}\label{sec:Graphs}
+\subsection{Graphs}\label{sec:graphs}
A Graph is a mathematical and informatical Concept, that allows you to store a more general Form of Data then just those encoded in Vectors. Namely, Graphs allows storing relational Information of an unbounded\footnote{For computional Reasons, Graphs are not completely unbounded in the following Chapters, but have a maximum size} amount of of Objects. This is done, by defining two Objects: Nodes, which are the Objects of Interest, and can be mathematically described by vectors\footnote{In theory you would not need to be able to define those Objects as Vectors, but for practical Application this is quite useful}, and Edges that are pairs of connected Node Indices, and thus encode the relation between those Nodes.
@@ -215,7 +217,7 @@ Given those relational applications, there are also applications that are not ba
%from file ..\..\write\/data\01intro\06graphae
-\subsection{Graph Autoencoder}\label{sec:Graph Autoencoder}
+\subsection{Graph Autoencoder}\label{sec:gae}
The main Problem of ParticleNet for finding new Physics is its supervised Approach. This means, that each new physics model can only be detected, if you train a special Network for it. Not only would this need a lot of Networks, with the corresponding high number of false positives, but this also limits there effectiveness, as you can only find new Physics that has already been suggested. This is why you could ask yourself if you could not combine the Graph Approach of ParticleNet with the Autoencoder Idea of QCDorWhat. This is the main Idea that is tried to Implement in this Thesis, and this Taks is definitely not trivial: Creating something like a Graph Autoencoder has some Problems, namely the fact, that a compression is usually not local\footnote{graph pooling operations are quite common, since the output of a graph network usually has a different format than its input. The way this is usually done, is by applying a function (mean,max for example) to each node. ParticleNet for example uses a GlobalAveragePooling (ENTER REFERENCE?), so it calculates the average over the nodes for each feature. This kind of pooling works quite well, but is sadly not really appliable to autoencoders, since those functions are not really invertible}. That does not mean, there are no Approaches, just that most Autors shy away from any Approach that chances the Graph Size\footnote{see (ENTER REFERENCE) or (ENTER REFERENCE)}. The first Paper you find, by just searching for a Graph Autoencoder, is a Paper by Kipf et Al (ENTER REFERENCE) and a lot of Paper referencing it. The main Problem here is, that Kipf et Al uses one fixed Adjacency Matrix, and thus one equal Graph Setup, for any Input. This allows for neither the learnable meaning of similarity that appearently makes ParticleNet so good, the variable inputsize discussed in (ENTER LINK) and, probably worst, not for any structural difference in any different Jets. Other Approaches come from the Problem of Graph Pooling Operations, meaning the definition of some kind of Layer, that takes a Graph as Input, and returns a smaller Graph in a learnable Manner\footnote{This is not an entirely solved Problem, but would be quite useful, since it allows for hirachical Learning, similar to the use of Pooling Layers in Convolutional Networks}. DiffPool (ENTER REFERENCE) and MinCutPool (ENTER REFERENCE) migth be good examples for this, but Graph U Nets (ENTER REFERENCE) stand out, since it also gives an Implementation to a anti Pooling layer, and thus allows for a Graph Autoencoder in the way we require it here , which is why the first Approach we tried is based on their Approach. See for this Chapter (ENTER CHAPTER).
@@ -228,12 +230,12 @@ The main Problem of ParticleNet for finding new Physics is its supervised Approa
%from file ..\..\write\/data\02tech\00intro
\newpage
-\section{Definitions}\label{sec:Definitions}
+\section{Definitions}\label{sec:secdef}
%from file ..\..\write\/data\02tech\01binclass
-\subsection{Binary Classification}\label{sec:Binary Classification}
+\subsection{Binary Classification}\label{sec:binclass}
The Task of evaluating finding the difference between Background and Signal Data has been studied a lot as a Boolean decision Problem. Notable use cases include (ENTER REFERENCE) and (ENTER REFERENCE). In general, they consider 4 fractions, the fraction of events that are of type background or signal, which are classified either as background and signal.
@@ -247,7 +249,7 @@ The Task of evaluating finding the difference between Background and Signal Data
-\subsubsection{ROC curve}\label{sec:ROC curve}
+\subsubsection{ROC curve}\label{sec:classroc}
For most decision Problems, these Fractions are a functions of some parameter. Consider the following output of an classifier:
\begin{figure}[H]
@@ -281,12 +283,12 @@ or, to focus on the Error Rate
-\subsubsection{Area Under the Curve}\label{sec:Area Under the Curve}
+\subsubsection{Area Under the Curve}\label{sec:classauc}
To simplify comparing ROC scores, you can use an AUC score to summarise it. This AUC score is defined as the integral of the true positive rate over the false positive rate. Obviously this is not perfect, since you reduce a function into only one number, but it is fairly wide accepted, and easy to interpret, as a perfect score would result in an AUC score of 1, while a Classifier that just guesses results in an AUC score of 0.5 and a perfect anticlassifier would result in 0. Also this reduction into only one number can makes the AUC score less errorprone than other values. 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 false negatives). This can result in Networks improving the AUC score by just chancing unimportant parts of the ROC curve.
-\subsubsection{other Measures}\label{sec:other Measures}
+\subsubsection{other Measures}\label{sec:classother}
This Problem of addressed focus is solved by the e30 measure. This measure corresponds to the inverse of the false positive rate corresponding to a true positive rate of 0.3. This means that the e30 score only cares about the accuracy at an acceptable true positive rate, but it also means, that this score is a bit more random, which is why it is not used very often in the following
@@ -295,20 +297,20 @@ This Problem of addressed focus is solved by the e30 measure. This measure corre
%from file ..\..\write\/data\02tech\02evmod
-\subsection{Problems in Evaluating a Model}\label{sec:Problems in Evaluating a Model}
+\subsection{Problems in Evaluating a Model}\label{sec:eval prob}
Even when we can evaluate a Binary Classification Problem\footnote{see Chapter (ENTER CHAPTER)}, this does not mean, that evaluating an Autoencoder is easy. This is a problem, since we basically want to do 2 Things at the same Time, creating an autoencoder and creating a classifier, and there are 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 useless.
-\subsubsection{AUC scores}\label{sec:AUC scores}
+\subsubsection{AUC scores}\label{sec:evalauc}
Your first Idea in how to evalutate an Autoencoder, migth be to simply use the Quality of the Classifier (the AUC Score, see Chapter (ENTER CHAPTER)), since the Classifier works 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 (ENTER CHAPTER multi model Plot AUC loss)), but in general this is simply not true, as Chapter (ENTER CHAPTER AUC through zero) shows. 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 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, and to assume that the AUC score correlates.
-\subsubsection{Losses}\label{sec:Losses}
+\subsubsection{Losses}\label{sec:evalloss}
So why not always use the loss: Look at the Quality of the Autoencoder and try to optimize only it. This again has Problems: Not only requires this still a strong relation between AUC and loss (That is not neccesarily given, consider the Problem of finding the best compression size: The loss will usually 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 (ENTER CHAPTER)}, which makes comparing different Networks only possible, if you neither alter the Loss nor the Normalization.
-\subsubsection{Images}\label{sec:Images}
+\subsubsection{Images}\label{sec:evalimg}
This cross comparison Problem, you can easily solve by simply looking at the Images behind the losses\footnote{the jet Image showing Input and Output of the Autoencoder, see for an example (ENTER CHAPTER)}. 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(see Chapter (ENTER CHAPTER)), 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, especcially since what problems you migth see in those images does not neccesarily correspond to what the network fails at. Most notably you seem to focus on angular differences, and mostly neglect differences in $lp_{T}$.
\begin{figure}[H]
@@ -321,7 +323,7 @@ This cross comparison Problem, you can easily solve by simply looking at the Ima
Sure, you can also look at the $p_{T}$ reproduction, but this demands weighting importance, and does not make evaluating images any easier.
-\subsubsection{Oneoff width}\label{sec:Oneoff width}
+\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 Chapters (ENTER CHAPTER) and (ENTER CHAPTER). Because of this, it will be explained in Chapter (ENTER CHAPTER). It seems to help with all the given Problems. There is a strong relation between this width and the AUC score\footnote{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.
@@ -330,7 +332,7 @@ Sure, you can also look at the $p_{T}$ reproduction, but this demands weighting
%from file ..\..\write\/data\02tech\04dataprep
-\subsection{Datapreperation}\label{sec:Datapreperation}
+\subsection{Datapreperation}\label{sec:data}
Most of the time (always except in Chapter (ENTER CHAPTER)), Jet Data provided by (ENTER REFERENCE) is used, mostly because it is easier to comapre their results. These Jets have a transverse momentum between $550 \cdot Gev$ and $650 \cdot Gev$ and a maximum Radius of $R \leq 0.8$.
@@ -364,10 +366,10 @@ You could ask yourself if it would not be a good idea to add more Variables. Par
%from file ..\..\write\/data\02tech\06explain
-\subsection{Explaining Graphics}\label{sec:Explaining Graphics}
+\subsection{Explaining Graphics}\label{sec:graphics}
-\subsubsection{Output Images}\label{sec:Output Images}
+\subsubsection{Output Images}\label{sec:imgout}
\begin{figure}[H]
\centering
@@ -388,7 +390,7 @@ In those Images, you see each Particles $\phi$ and $\eta$, including any normali
This way of looking at the Network performance is quite useful for finding patterns in the data. There seem to be Networks that show a high correlation between the angles(see (ENTER CHAPTER)), and it is quite common for the reproduced Values to have less spread than the input one (see (ENTER CHAPTER)). A problem here is, that you can only look at some Images, and finding one good reproduction for each Network is not that hard. To compat this, we use always the same event\footnote{the same event for each training set to be precise}.
-\subsection{AUC Feature Maps}\label{sec:AUC Feature Maps}
+\subsubsection{AUC Feature Maps}\label{sec:imgmaps}
\begin{figure}[H]
\centering
@@ -401,7 +403,7 @@ This way of looking at the Network performance is quite useful for finding patte
Since a L2 loss is just a mean over a lot of L2 losses for each Feature and Particle, you could use the original losses, to also get an AUC score for each Feature and Particle. These AUC scores are shown in Feature Maps, showing the Quality of each combination of Feature on the horizontal axis and Particle on the verticle axis in the form of pixels. A perfect classifier($AUC = 1$) would result in a dark blue pixel, a perfect anti classifier($\operatorname{EQ}{\left(AUC,0 \right)}$) would be represented by a dark red pixel, and a useless classifier, that guesses or always results in the same class($AUC = 1 / 2$) would be a white classifier. In short: The more colorful a pixel is, the better it is, and an Autoencoder trained on qcd events should be blue, while an Autoencoder trained on top events has to be red.
The nice thing about those maps, is that they can show the focus of the Network, as well as its problems. Since a perfect reconstruction has no decicion power, the same as a terrible reconstruction, a Network that has focus problems\footnote{reconstructs some things much better than other things, making both parts worse}, can be clearly seen in those maps (see (ENTER CHAPTER)). Also it is fairly common, to get one feature and Particle, that has more decision power, than the whole combined Network (see (ENTER CHAPTER) for an example and (ENTER CHAPTER) for the explanation). Finally, an AUC map that is completely blue or red is quite uncommon, more probably some Features are red, some are blue, and you get an indication on which Features are useful for the current Task (see (ENTER CHAPTER)).
-\subsection{Network Setups}\label{sec:Network Setups}
+\subsubsection{Network Setups}\label{sec:imgsetups}
\begin{figure}[H]
\centering
@@ -442,7 +444,7 @@ As you see, this is a relatively easy Graph Autoencoder, but explaining each lay
%from file ..\..\write\/data\02tech\09graphnet
-\subsection{Graph Neuronal Networks}\label{sec:Graph Neuronal Networks}
+\subsection{Graph Neuronal Networks}\label{sec:gnn}
Graph Neuronal Networks are defined by a Graph Update Layer. This Layer takes all Feature Vectors of some graph, aswell as their corresponding Graph Connections to return an updated Feature Vector. To do this, this Layer is build from two different Interactions, the Update step of each node itself, which is call the self Interaction Term here, and the Update step of a node corresponding to its neighbouring Nodes in the Graph. This is call the neighbour Interaction Term.
@@ -468,12 +470,12 @@ So the activation reduces to two constants $a$ and $b$, and in Practice this rel
%from file ..\..\write\/data\03graphae\00title
\newpage
-\section{Making Graph Autoencoder work}\label{sec:Making Graph Autoencoder work}
+\section{Making Graph Autoencoder work}\label{sec:secgae}
%from file ..\..\write\/data\03graphae\01failed
-\subsection{Failed Approaches}\label{sec:Failed Approaches}
+\subsection{Failed Approaches}\label{sec:failed}
@@ -481,13 +483,13 @@ So the activation reduces to two constants $a$ and $b$, and in Practice this rel
In this Chapter we will quickly go over some Ideas you could have, on how to implement a graph autoencoder and finish with the first implementation that could be considered working.
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 (ENTER CHAPTERLINK).
-\subsubsection{trivial models}\label{sec:trivial models}
+\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 still 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 (ENTER CHAPTER)). 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.
That beeing said, permutation Invariance can also be a benefit, especially in Permutation Invariant Input data, more to this in Chapter (ENTER CHAPTERLINK)
-\subsubsection{minimal models}\label{sec:minimal models}
+\subsubsection{minimal models}\label{sec:failedminimal}
To improve this model, we started working with smaller Graph sizes (mostly the first 4 particles), making the structure less complicated, and allowing for more experimentation thanks to the lower time cost. Notable improvements include replacing the added zeros by a learnable Function of the remaining parameters, relearning the graph on the new parameterspace and adding some Dense Layers after the Graph Interactions, but the most important Improvment was achieved by making the compression and decompression local in some learning axis. Instead of just removing parameters in an arbitrary way of physical intuition, we demand that particles which are similar in some way are to be compressed together: This is achieved by the creation of a Function that compresses a set of particles into one particle, and allow the Network to learn what similarity means\footnote{In the compression step, we define a new Feature for each node, by which we sort the set of nodes, and afterwards we build sets of n particles from this ordering, and compress them using a linear function (it migth be interresting to look at nonlinear functions, but we generally see worse results by adding an alinearity). Please note that since we use a Feature to sort the elements, and in the Graph Update step there are neighbour steps, that generally increase similarity, connected particles are more probably compressed together.}.
These Networks still have problems, as we will discuss in the following, but generally produce respectable Decision Qualities, and show ("sometimes", see Chapter (ENTER CHAPTERLINK)) similarities between Input and Output Image. These Network is discussed in the next subchapter.
@@ -497,7 +499,7 @@ These Networks still have problems, as we will discuss in the following, but gen
%from file ..\..\write\/data\03graphae\02simpleae
-\subsection{An explicit look at the first working graph autoencoder}\label{sec:An explicit look at the first working graph autoencoder}
+\subsection{An explicit look at the first working graph autoencoder}\label{sec:firstworking}
@@ -513,14 +515,14 @@ These Networks still have problems, as we will discuss in the following, but gen
As you see, this Autoencoder takes qcd jets, transforms them as introduced in (ENTER CHAPTER), does some additional preprocessing, after which a graph is constructed, an graph update step is run, after which a graph compression algorithm is applied just to be reversed afterwards, to reconstruct a new graph, which is again used to update the feature vector, after which (and after a sorter), the current graph is compared. There are some Layers, that have not explicitely been explained before, so this is what we want to do in this Chapter
-\subsubsection{topK}\label{sec:topK}
+\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 solved by demanding that the metrik is entirely diagonal, 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 (ENTER APPENDIX)).
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 (ENTER APPENDIX).
Finally, it should be noted, that the topK layer can increase the size of the Feature vectors, which is useful for the compression algorithm, even though in this specific example this is not used.
-\subsubsection{Compression}\label{sec:Compression}
+\subsubsection{Compression}\label{sec:quickcompression}
Also this Layer is defined by a learnable sense of locality, but instead of defining a metrik, we simply use the last parameter of the input feature vectors\footnote{This is essentially the same, since it is preceded by a graph update layer, which redefines each Feature with its self interaction matrix, but it has one definite benefit, as the neigbour interaction matrix, tends to average graph connected feature vectors, and thus two nodes, that are connected in the adjacency matrix, are more similar} to sort each feature vector, to split the list of feature vectors in lists of feature vectors, that are compressed together\footnote{in this example, the 4 input features are compressed by sorting into sets of 4 feature vectors, so you could ask yourself if it actually does something to sort by a learnable sense of locality for minimal models, but they at least help to keep the permutation invariance}, which are then transformed using a simple dense layer into a new feature vector each.
\begin{figure}[H]
@@ -534,7 +536,7 @@ Also this Layer is defined by a learnable sense of locality, but instead of defi
After this Graph compression, which reduces $4$ times $flag + 3$ parameters into one vector with $10$ parameters, we use 3 dense layers reducing the parameter count down to $6$
-\subsubsection{Decompression}\label{sec:Decompression}
+\subsubsection{Decompression}\label{sec:quickdecompression}
After inverting the Dense Layers, we again have a compressed Vector with 10 Features.
@@ -548,12 +550,12 @@ As each list of vectors is compressed into only one feature vector, the inverse
-\subsubsection{Sorting}\label{sec:Sorting}
+\subsubsection{Sorting}\label{sec:quicksort}
Since the compression stage reordered each feature vector, while the decompression algorithm does not reorder, comparing lists of vectors is not that easy. It should be clear, that a vector that may be perfectly reconstructed, but shuffled in a random way, can have an neirly arbitrarily big loss. This is solved by the initial and final sorting. These layers simply order each feature vector by its $lp_{T}$ Value, so that at least perfectly reconstructed feature vectors are compared the rigth way. It should be noted, that this sorting is not strictly neccesary, as the Network will learn to work even when you just compare vectors in the way, the nodes are given, but sorting makes this task easier, allowing the Network to focus on more important things, and thus generally working better\footnote{The central problem migth be, that the sorting kind of breaks graph permutation symmetry}. See Chapter (ENTER CHAPTER) for an experimental comparison on a more advanced graph autoencoder.
-\subsubsection{Training Setup}\label{sec:Training Setup}
+\subsubsection{Training Setup}\label{sec: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
\begin{figure}[H]
@@ -567,7 +569,7 @@ Another thing that has to be clearified concerning this model, is the training p
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 (ENTER CHAPTER)), and you could even ask yourself if it would not be possible to remove the whole need if 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 errorbased 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:Results}
+\subsubsection{Results}\label{sec:quickres1}
So why do we consider this the first working model?
This migth be the first model, that can show some resemblence between the Input and the Output
@@ -607,7 +609,7 @@ If we look at the Featuremap, the first thing you should notice is the perfect S
%from file ..\..\write\/data\03graphae\03goodae
-\subsection{Improving Autoencoder}\label{sec:Improving Autoencoder}
+\subsection{Improving Autoencoder}\label{sec:secondworking}
This good AUC score, looks like to only thing we now need to do, is to increase the size, and we probably have an Autoencoder that can rival the best other Autoencoder. But before we try, and fail\footnote{see Chapter (ENTER CHAPTER)}, 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.
@@ -621,11 +623,11 @@ This good AUC score, looks like to only thing we now need to do, is to increase
-\subsubsection{Training Setup}\label{sec:Training Setup}
+\subsubsection{Training Setup}\label{sec:quick2setup}
Here we still train on the first $4$ nodes, and with the same batch size of $200$, but with a lower learning rate of $0.0003$ and with a higher patience, stopping only after the network do not improve for 30 epochs. We also increase the compression size from 6 to 7.
-\subsubsection{Results}\label{sec:Results}
+\subsubsection{Results}\label{sec:quick2results}
\begin{figure}[H]
@@ -674,29 +676,167 @@ Finally, the AUC Feature map is way better than before: The angular collumns are
%from file ..\..\write\/data\03graphae\03optimalae
-\subsection{Making Autoencoder good}\label{sec:Making Autoencoder good}
+\subsection{Making Autoencoder good}\label{sec:thirdworking}
-\subsubsection{better encoding}\label{sec:better encoding}
+\subsubsection{better encoding}\label{sec:encoding}
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 adding 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 the global matrix. This is done here, by applying a function to each submatrix. Here, 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.
-(ENTER RESULTS)
+\begin{table}[h!]
+\centering
+\begin{tabular}{|c | c c c c c|}
+
+\hline
+ & mean loss & loss std & n & auc & oneoff auc \\
+
+\hline
+
+Rounded Min & $0.366$ & $0.037$ & $5$ & $0.533$ & $0.623$ \\
+
+Rounded Max & $0.388$ & $0.042$ & $6$ & $0.563$ & $0.484$ \\
+
+Rounded Mean & $0.342$ & $0.036$ & $5$ & $0.56$ & $0.556$ \\
+
+\hline
+Min & $0.371$ & $0.025$ & $6$ & $0.559$ & $0.562$ \\
+
+Max & $0.372$ & $0.025$ & $6$ & $0.565$ & $0.541$ \\
+
+Mean & $0.351$ & $0.04$ & $5$ & $0.54$ & $0.486$ \\
+
+
+\hline
+
+\end{tabular}
+\caption{Quality differences for different encoder with a learnable handling of the Feature vectors}
+\label{table:encode1}
+\end{table}
+
+
+
Also we test, how well a learnable parameter transformation, as used before, works, compared to also applying a function (mean, max, min) to each feature vector
(ENTER RESULTS)
-\subsubsection{better decoding}\label{sec:better decoding}
+\begin{table}[h!]
+\centering
+\begin{tabular}{|c | c |c c c c c|}
+
+\hline
+ & function & mean loss & loss std & n & auc & oneoff auc \\
+
+\hline
+
+Rounded Min & mean & $0.366$ & $0.016$ & $4$ & $0.656$ & $0.472$ \\
+
+Rounded Max & mean & $0.333$ & $-1$ & $1$ & $0.656$ & $0.549$ \\
+
+Rounded Mean & mean & $0.372$ & $0.016$ & $4$ & $0.656$ & $0.542$ \\
+
+\hline
+Min & mean & $0.37$ & $0.007$ & $4$ & $0.656$ & $0.503$ \\
+
+Max & mean & $0.362$ & $0.002$ & $3$ & $0.654$ & $0.55$ \\
+
+Mean & mean & $0.363$ & $0.01$ & $4$ & $0.655$ & $0.505$ \\
+
+\hline
+Rounded Mean & min & $0.296$ & $0.022$ & $5$ & $0.579$ & $0.543$ \\
+
+Rounded Mean & max & $-1$ & $-1$ & $-1$ & $-1$ & $-1$ \\
+
+
+\hline
+
+\end{tabular}
+\caption{Quality differences for different encoder with a fixed function}
+\label{table:encode2}
+\end{table}
+\subsubsection{better decoding}\label{sec: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 (ENTER CHAPTER) 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 learnabl 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 (ENTER CHAPTER) 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.
(ENTER RESULTS)
+
+
+\begin{table}[h!]
+\centering
+\begin{tabular}{|c | c c c c c|}
+
+\hline
+ & mean loss & loss std & n & auc & oneoff auc \\
+
+\hline
+
+Product & $0.265$ & $0.019$ & $4$ & $0.568$ & $0.557$ \\
+
+Sum & $0.305$ & $0.026$ & $5$ & $0.566$ & $0.514$ \\
+
+Or & $0.404$ & $0.248$ & $18$ & $0.562$ & $0.502$ \\
+
+And & $0.354$ & $0.074$ & $22$ & $0.587$ & $-1$ \\
+
+
+\hline
+
+\end{tabular}
+\caption{Quality differences for different graph like decoder}
+\label{table:decode1}
+\end{table}
+
+
We can also look at the way, the original Graph is combined with the newly generated Graph. Instead of using a product, we can also use a sum, or again round the result to an or\footnote{In Practice we expect the product to be virtually identical to the or, since the inputs are either 1 or 0} or an and. Since the combination with a constant graph is not very interresting, we use paramlike decompression for the practical results:
(ENTER RESULTS)
+\begin{table}[h!]
+\centering
+\begin{tabular}{|c | c c c c c|}
+
+\hline
+ & mean loss & loss std & n & auc & oneoff auc \\
+
+\hline
+
+Product & $0.28$ & $0.037$ & $4$ & $0.553$ & $0.522$ \\
+
+Sum & $0.3$ & $0.024$ & $4$ & $0.549$ & $0.526$ \\
+
+Or & $-1$ & $-1$ & $10$ & $-1$ & $-1$ \\
+
+And & $0.348$ & $0.038$ & $16$ & $0.631$ & $0.546$ \\
+
+
+\hline
+
+\end{tabular}
+\caption{Quality differences for different param like decoder}
+\label{table:decode2}
+\end{table}
Finally, since we now have a little Adjacency Matrix, and a list of Feature vectors for each original Feature vector, we can apply a Graph Update Step on those subgraphs, to hopefully enhance the decompression by mixing the decompression of the Adjacency matrix and the Feature vectors. This was already enabled in all previous compression and decompression tests, but is tested here again on paramlike decompression:
(ENTER RESULTS)
+\begin{table}[h!]
+\centering
+\begin{tabular}{|c | c c c c c|}
+
+\hline
+ & mean loss & loss std & n & auc & oneoff auc \\
+
+\hline
+
+yes & $0.28$ & $0.037$ & $4$ & $0.553$ & $0.522$ \\
+
+no & $0.277$ & $0.0335$ & $3$ & $0.571$ & $0.528$ \\
+
+
+\hline
+
+\end{tabular}
+\caption{Quality difference for either running learnable sub graph updates or not}
+\label{table:decode3}
+\end{table}
+
%from file ..\..\write\/data\03graphae\04identityproblems
-\subsection{Building Identities out of Graphs}\label{sec:Building Identities out of Graphs}
+\subsection{Building Identities out of Graphs}\label{sec:identities}
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 (ENTER CHAPTER), the Graph Update Step is given by
@@ -727,7 +867,7 @@ WRITE SOMETHING ABOUT OTHER IDENTITY PROBLEMS, namely permutation invariance (se
%from file ..\..\write\/data\03graphae\04scaling
-\subsection{Scaling}\label{sec:Scaling}
+\subsection{Scaling}\label{sec:scaling}
NOT AT THE RIGTH POSITION
@@ -744,7 +884,7 @@ Given working small models, you migth think, that creating a bigger model is fai
Why is that? C addition (see Chapter (ENTER CHAPTER)): The Network focusses on each Part of the Jet the same, but in the Particles with higher $p_{T}$ there is more Information, that gets watered down, by adding less accurate Information to it.
-\subsubsection{Combination for Scaling}\label{sec:Combination for Scaling}
+\subsubsection{Combination for Scaling}\label{sec:scalcomb}
The easiest way to understand this, is by splitting up the Network into small Networks. Here we use $n$ $4$ Particle Networks, instead of one $4 \cdot n$ Particle Network. Given the sum of those Networks, Scaling is still a Problem
\begin{figure}[H]
@@ -765,7 +905,7 @@ but we now can use C addition to fix the focus of the Network: We can approximat
-\subsubsection{Redefining losses}\label{sec:Redefining losses}
+\subsubsection{Redefining losses}\label{sec:scalloss}
One thing to remember here, is that these Networks are still just a combination of trivial Models. So the obvious question is how to apply this to bigger Networks? The first Idea migth be to redefine the loss, in a way, that the loss of any particle get multiplied by the factors used in the splitted Networks. The Problem here is that this chances the focus of the Autoencoder again: Since then it can make more errors in the later particles, it will, making the later Particles less useful for classification, but also making the first Particles more useless, since the focus is now lying on them, making their reconstruction better, up to a point at which their reconstruction is to good to find any difference between Background and Signal.
Another Idea migth be to just apply this loss weigthing in the Evaluation Phase, and not in Training. This definitely helps, but in my tries does not seem to be enough, since the same effect as before works now in the opposite way: Particles with lower $p_{T}$ have a higher inaccuracy, that translates to a higher loss for them, and the Autoencoder focussing more on them, making the lower Particles and the higher Particles less useful.
@@ -783,12 +923,12 @@ That does not mean that a good scaled Network is impossible, the Ideas from Chap
%from file ..\..\write\/data\03graphae\05losses
-\subsection{Losses}\label{sec:Losses}
+\subsection{Losses}\label{sec:losses}
Choosing what the Network focusses on, is done through two things: The Inital Normalisation, and the Loss Function. While the Size of the Initial Variables is relatively straigth forward\footnote{The bigger Each Value, the bigger its contribution to the loss}, choosing different loss functions makes less predictable chances, so I will discuss some different Methods here
-\subsubsection{L2}\label{sec:L2}
+\subsubsection{L2}\label{sec:l2}
Beeing probably the most obvious choice, setting the loss function to be the quadratic difference between Input and Output still has some benefits
\begin{equation}loss = \left(inn - out\right)^{2}\end{equation}
@@ -802,7 +942,7 @@ Not only is this easy to implement and fast to compute, but it also punishes big
-\subsubsection{Evalutating losses}\label{sec:Evalutating losses}
+\subsubsection{Evalutating losses}\label{sec:evalloss}
To evaluate the Quality of a loss concerning this Problem, we define a new Parameter $varfrac$ as follows:
\begin{equation}varfrac = \operatorname{min}{\left(\frac{\operatorname{std}{\left(A^{i}{\left(x \right)} \right)}}{\operatorname{std}{\left(x^{i} \right)}} \right)}\end{equation}
@@ -811,7 +951,7 @@ This Value is generally between 0 and 1. A Value of $0$ would mean, that the Net
-\subsubsection{Ln}\label{sec:Ln}
+\subsubsection{Ln}\label{sec:ln}
The first other kind of loss you can look at, would be a Ln loss. For each $\operatorname{LesssThan}{\left(2,n \right)}$, this Loss still has the same Problem of uncertain Losses, but for smaller n it does not. L1 does not prefer lower Predictions\footnote{the loss described above would now look like $a \cdot \alpha + a \cdot \left(1 - \alpha\right) - b + b = a + b \cdot \left(1 - 2 \cdot \alpha\right)$ which is minimal (for each $0.5 \leq \alpha$ for b beeing as big as possible (this loss assumes $b \leq a$, and anything Guessing rigth less than half of the Time, would still result in the Network learning not to guess, as we would want}, and a n lower than 1 would reverse the Effect entirely\footnote{We tried $n = 1 / 2$, but this results in NANs, so we had to tweak $\sqrt{\operatorname{abs}{\left(a - b \right)}}$ into $\sqrt{\operatorname{abs}{\left(a - b \right)} + 1} - 1$ since the NANs are a result of the square root not beeing differentiable at 0. Using this Tweak, the same loss looks like $\alpha \cdot \sqrt{a - b + 1} + \left(1 - \alpha\right) \cdot \sqrt{a + b + 1} - 2$ (ENTER MINIMIZATION)}. And in a sense, this works, those Networks have the same Input width as Output width, but the different loss has another effect: Since now one big loss is at most as worse as two small ones, Networks, that remember some Values exactly, while guessing the remaining ones, are common.
\begin{figure}[H]
@@ -834,7 +974,7 @@ You could ask yourself if this is even a problem. Since these Networks are not j
-\subsubsection{Image like losses}\label{sec:Image like losses}
+\subsubsection{Image like losses}\label{sec:imageloss}
One thing to consider, is that Image like Networks usually do not have these Problems. The main difference in losses, is the fact, that Imagelike Networks do not compare all Values to each other, but only one Value, if the other Values match. A pixel is restored with zero loss, if its angles are about correct, and the Momentum is correct. If the angles are a bit to much off, the loss is derived by comparing the Input to the reconstructed zero, and the newly constructed Momentum to the zero of a pixel that was initialy zero. This means, that each difference in angles, is punished the same, and the only way, for the Network to improve, is to guess those rigth and if we could implement this here, this would solve the Problem of guessing to low, at least in angles\footnote{The Problem stays the same for $p_{T}$, since guessing wrongly zero is punished way less, than guessing wrongly $a$. So for the same setup as before, the new loss looks exactly the same as in the Ln case $\alpha \cdot \left(a - b\right)^{n} + \left(1 - \alpha\right) \cdot \left(a + b\right)^{n}$}. So how to implement this: The first Idea, would be to set the $p_{T}$ reconstruction to zero, if the angles are more than a certain difference apart. This works fine, but offers no control and for implementation purposes, a symmetric loss function would be nice. We choose the following extension
\begin{equation}\left(1 - f{\left(d \right)}\right) \cdot c{\left(x \right)} + \left(p_{T}^{a} - p_{T}^{b}\right) \cdot f{\left(d \right)}\end{equation}
@@ -855,7 +995,7 @@ From my (limited) experimentation, choosing a continous function $f{\left(d \rig
%from file ..\..\write\/data\03graphae\06caddition
-\subsection{C addition}\label{sec:C addition}
+\subsection{C addition}\label{sec:caddition}
@@ -926,7 +1066,7 @@ You migth seem some sligth deviations from the expected result. These migth be e
%from file ..\..\write\/data\03graphae\08whygae
-\subsection{Why use Graph Autoencoder}\label{sec:Why use Graph Autoencoder}
+\subsection{Why use Graph Autoencoder}\label{sec:why gae}
NOT AT THE RIGTH POSITION
@@ -946,13 +1086,13 @@ MAYBE SOMETHING ABOUT VARIABLE MEANING OF LOCALITY
%from file ..\..\write\/data\04aucsinv\01intro
\newpage
-\section{Auc Scores and Invertibility}\label{sec:Auc Scores and Invertibility}
+\section{Auc Scores and Invertibility}\label{sec:secinv}
%from file ..\..\write\/data\04aucsinv\02simplicity
-\subsection{alternative models}\label{sec:alternative models}
+\subsection{Simplicity}\label{sec:simplicity}
NOT THE BEST POS WS
@@ -1016,7 +1156,7 @@ Since I tried to train a Model, that was a great seperator, when I chanced my in
%from file ..\..\write\/data\04aucsinv\03inv
-\subsection{Problems in Invertibility}\label{sec:Problems in Invertibility}
+\subsection{Problems in Invertibility}\label{sec:inv}
Given a Classifier that if trained on qcd, finds top jets different, you could (and should) ask yourself if this is just a feature of your data, as shown in Chapter (ENTER CHAPTER) or if this is more general. The easiest way to test this, is just to swithc the meaning of signal and background: Can an Autoencoder trained on top jets classify qcd jets as different. To keep those plots easily readable, I keep qcd as background, which results in an AUC score of 0 beeing optimal for those switched Networks, but when you look at a simple model, it is not
@@ -1034,17 +1174,17 @@ All of this is obviously quite problematic, and thus will be solved in the very
%from file ..\..\write\/data\04aucsinv\04norm
-\subsection{solving Invertibility through Normation}\label{sec:solving Invertibility through Normation}
+\subsection{solving Invertibility through Normation}\label{sec:normation}
-\subsubsection{The Meaning of Complexity}\label{sec:The Meaning of Complexity}
+\subsubsection{The Meaning of Complexity}\label{sec:complexity}
Since there is a Model, that can reach a similar quality as an good Autoencoder, by just comparing the Angular Data to zero, it stands to reason, that these Autoencoder work in a similar way, which of course would not make them Invertible\footnote{a model comparing just angles to zero, would be the same, if trained on qcd or on top jets}.
We can test this Hypothesis, by just removing any kind of Data difference, that allows for this trivial comparison, which, as shown in Chapter (ENTER CHAPTER) is the width of the Input Distribution, or in other words: What we need is some kind of Normation.
This method has one obvious drawback: Not only do we actively remove Information, which will hurt the Performance, but we remove the Information, that generates most of the quality. This means, even if the resulting Classifier is invertible, it will probably be way worse than the trivial one.
-\subsubsection{How to normalise an Autoencoder}\label{sec:How to normalise an Autoencoder}
+\subsubsection{How to normalise an Autoencoder}\label{sec:normprobs}
One thing, I did not realise after triying to normalise my Input Datapoints, is that simply demanding that the mean is zero and the standart deviation is one, does not work. This migth be an effect that is most important when we talk about small Networks\footnote{Networks with a low amount of Input Particles}, but is still somewhat of an effect in every Network, and becomes important in Chapter (ENTER CHAPTER ONEOFF). The Problem is, that by demanding a value to be fixed, we remove one information from 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\footnote{3, since there are 3 Variables which mean and 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\footnote{ignoring flag for now, three Values is always enough to encode 4 flag Values, since the first three flag Values are always one (the jet with the lowest number of particles has 3???? particles in my trainingsset)}. In reality things are not so easy. There is no garantee that this minima is found, and the Graph Structure does not neccesarily allow for this kind of transformation(see Chapter (ENTER CHAPTER)), but training this kind of Network, does not result in the Model gaining any Classification Power. This can be seen in the corresponding Feature Map
\begin{figure}[H]
@@ -1099,7 +1239,7 @@ $z{\left(y \right)}$ is scale Invariant, Scale Invariance is also given\footnote
-\subsubsection{Using this normation}\label{sec:Using this normation}
+\subsubsection{Using this normation}\label{sec:usenorm}
Using this kind of Normation, at least 4 node Networks are invertible. And not only this, but also each Feature is Invertible.
\begin{figure}[H]
@@ -1163,7 +1303,7 @@ training for 1000 epochs, and then until the loss increases for 250 epochs, resu
-\subsubsection{Improving the Normation even further}\label{sec:Improving the Normation even further}
+\subsubsection{Improving the Normation even further}\label{sec:normplus}
After seeing what an effect some kind of Normation can have, we are not completely satisfied anymore with the normated Feature Maps:
\begin{figure}[H]
@@ -1204,7 +1344,7 @@ This I will explain in Chapter (ENTER CHAPTER).
%from file ..\..\write\/data\04aucsinv\06aucs
-\subsection{Auc Scores for Toptagging}\label{sec:Auc Scores for Toptagging}
+\subsection{Auc Scores for Toptagging}\label{sec:aucfortt}
HAVE I ALREADE WRITTEN THIS?
@@ -1235,13 +1375,13 @@ SEE ABOVE
%from file ..\..\write\/data\05otherdata\01intro
\newpage
-\section{Data}\label{sec:Data}
+\section{Data}\label{sec:secdata}
%from file ..\..\write\/data\05otherdata\02ldm
-\subsection{Ligth Dark Matter}\label{sec:Ligth Dark Matter}
+\subsection{Ligth Dark Matter}\label{sec:ldm}
this chapter is missing, since it is still beeing worked on
@@ -1249,7 +1389,7 @@ this chapter is missing, since it is still beeing worked on
%from file ..\..\write\/data\05otherdata\03quarkgluon
-\subsection{Quark v Gluon}\label{sec:Quark v Gluon}
+\subsection{Quark v Gluon}\label{sec:qg}
this chapter is missing, since it is still beeing worked on
@@ -1260,13 +1400,13 @@ this chapter is missing, since it is still beeing worked on
%from file ..\..\write\/data\06othernets\01title
\newpage
-\section{Other Approaches}\label{sec:Other Approaches}
+\section{Other Approaches}\label{sec:secother}
%from file ..\..\write\/data\06othernets\03oneoff
-\subsection{Oneoff Networks}\label{sec:Oneoff Networks}
+\subsection{Oneoff Networks}\label{sec:oneoff}
When you consider the following Feature Map of a well normated Network:
@@ -1282,7 +1422,7 @@ You migth notice, that most of the decision power is in the first feature, but t
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 compressed 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 has no physical Meaning. More likely is the following: The Network saves one Value in the Compression Space, and is able to reconstruct an one from all the other Parameters. This makes sence, since we got this Auc Distribution by chancing the Normation 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 thus by chancing the Inputs, the constant is probably 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 on Paper it seams like a great Idea: As shown before (see Chapter (ENTER CHAPTER)), complexity is to a big part just width. You may be able to solve this by Normation, 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 normation 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. This results in very high correlations in the Outputs, but seems to simplify the convergence} 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 can result in the Network beeing able to learn more complicated functions.
-\subsubsection{oneoff math}\label{sec:oneoff math}
+\subsubsection{oneoff math}\label{sec:oomath}
Before we talk about the results, lets talk about the math 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 (ENTER LINK), and it is true, that by considering how to combine two Background, the Math is the 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 (ENTER LINK). This migth suggest, that the resulting combination of an OneOff Network is the combination with the highest possible AUC. Sadly this is simply not true, so where did this go wrong: The Problem are the assumptions made in C addition: 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 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 distance could be some sort of signal Data, that you want to catch. 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 focus width a width $s_{2} \leq s_{1}$: Which Peak is more probable to seperate any signal from the Background? Since the second Peak is less wide, it is less probable for a Signal Peak to overlap it, than to overlap the bigger Peak, and thus the better optimized Peak, 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 chances 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 is useful}
@@ -1304,7 +1444,7 @@ THE ABOVE MATH STILL NEEDS SOME DOUBLECHECKING
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 readjust 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 they are not always the same. In the following mostly readjusted means are used.
-\subsubsection{oneoff quality}\label{sec:oneoff quality}
+\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.
@@ -1321,7 +1461,7 @@ Now for Numbers: A simple oneoff Network reaches usually an Auc of about $0.6$,
Sadly, this observation is not really useful, since stopping the Training at the optimal Epoch would not be unsupervised, but it is quite interresting, since it shows, that there is some potential in those kind of networks, which is just not utilised good enough\footnote{this will be solved in Chapter (ENTER CHAPTER)}.
Another Problem is again invertibility: It is possible to create an invertible oneoff Network, but it is not trivially given. They work best, when you use a certain minimal complexity. To do this, a Graph Network seems to be less useful, than just a simple dense Network.
-\subsubsection{oneoff outside of physics}\label{sec:oneoff outside of physics}
+\subsubsection{oneoff outside of physics}\label{sec: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\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 (ENTER REFERENCE)}. 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 my results in the Appendix (ENTER APPENDIX), 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 (ENTER CHAPTER)) 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]
@@ -1342,12 +1482,12 @@ This apparently unused Potential led us to try them out on more classical evalua
%from file ..\..\write\/data\06othernets\05otheralgo
-\subsection{Other Algorithms}\label{sec:Other Algorithms}
+\subsection{Other Algorithms}\label{sec: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.
-\subsubsection{Support Vector Machines}\label{sec:Support Vector Machines}
+\subsubsection{Support Vector Machines}\label{sec: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, as it keeps the SVM from overfitting, but it also only allows it to learn certain distributions, so distributions like
\begin{figure}[H]
@@ -1360,11 +1500,11 @@ 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\footnote{implementent in sklearn} on qcd jets to find top jets does not result in any useful results (an AUC of ???).
-\subsubsection{k neirest neighbours}\label{sec:k neirest neighbours}
+\subsubsection{k neirest neighbours}\label{sec:whatsnvm}
Another usually quite useful algoritm is also an extension of the 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(ENTER IMAGE), with an quite good AUC of ????\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.
-\subsubsection{Isolation Forests}\label{sec:Isolation Forests}
+\subsubsection{Isolation Forests}\label{sec:whatsiforest}
An Isolation Forest\footnote{(ENTER REFERENCE)} 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 classifiyng 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 ???.
@@ -1377,12 +1517,12 @@ An Isolation Forest\footnote{(ENTER REFERENCE)} works quite different to the Alg
%from file ..\..\write\/data\07oneoffplus\00intro
\newpage
-\section{Mixed Approaches}\label{sec:Mixed Approaches}
+\section{Mixed Approaches}\label{sec:secmixed}
%from file ..\..\write\/data\07oneoffplus\01idea
-\subsection{Compressed One Class Learning}\label{sec:Compressed One Class Learning}
+\subsection{Compressed One Class Learning}\label{sec:mixedidea}
The main Problem of Autoencodern is the fact that its loss function is not neccesarily the best seperator\footnote{see Chapter (ENTER CHAPTER) and CHAPTER (ENTER CHAPTER)}, while the Problem of other Methods is the fact that they are not well equiped to handle symmetries of certain kind or trivial Information\footnote{In the case of oneoffs}, so maybe combining both Methods is not a bad idea: You train an Autoencoder just to convert the Input Space into the compressed Space, to run other algorithms on this compressed Space. This means that the seperator is now usually quite good, and that the Autoencoder can filter out symmetries and trivial Inputs. This Idea is not neccesarily new\footnote{see (ENTER REFERENCE)} and becomes harder to train, since both Networks have to be trained well, but this usually works quite well.
@@ -1395,7 +1535,7 @@ One Question you could ask yourself, if you want to train your Autoencoder on th
%from file ..\..\write\/data\07oneoffplus\02diffalgo
-\subsection{Different Algorithms}\label{sec:Different Algorithms}
+\subsection{}\label{sec:mixedalt}
Here I test different One Class Algorithms on one Autoencoder that trains on the first 4 Particles of top jets and tries to find qcd jets as a signal. For comparison, simply looking at the loss of this Network, you get an AUC score of about $0.377$ and the best AUC score I have seen for an Autoencoder is about $0.25$. I choose this one, since its reconstruction seems to be quite accurate, meaning that most information about the Jet is still contained in the Feature Space
@@ -1408,21 +1548,21 @@ Here I test different One Class Algorithms on one Autoencoder that trains on the
-\subsubsection{SVM}\label{sec:SVM}
+\subsubsection{SVM}\label{sec:mixedSVM}
A SVM works better on the compressed Space, now reaching an AUC of $0.434$, but even though this is definitely an improvement, it is still worse than just using the Autoencoder
-\subsubsection{Isolation Forest}\label{sec:Isolation Forest}
+\subsubsection{Isolation Forest}\label{sec:mixediforest}
Also the Isolation Forest improves, now reaching an AUC of $0.377$, so it is at least as good as simply using the Autoencoder.
-\subsubsection{k neirest neighbour}\label{sec:k neirest neighbour}
+\subsubsection{k neirest neighbour}\label{sec:mixedknn}
K neirest neighbour is the first algorithm that improves over simply using the Autoencoder loss, reaching an AUC of $0.307$, even though it should be noted, that this is for $k = 1$ and gets worse for higher $k$. Also this is still worse than our best Autoencoder.
-\subsubsection{oneoff}\label{sec:oneoff}
+\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 my autoencoder, and by combining multiple ones, you can reach an AUC of about $0.2$ beeing quite good.
@@ -1447,12 +1587,12 @@ This Chapter is missing since it is still beeing worked on
%from file ..\..\write\/data\07oneoffplus\08whyaesuck
-\subsection{Why Autoencoder might not be so great}\label{sec:Why Autoencoder might not be so great}
+\subsection{Why Autoencoder might not be so great}\label{sec:whynotgae}
NOT AT THE RIGTH POSITION
-\subsubsection{Reproduding vs Classifing Quality}\label{sec:Reproduding vs Classifing Quality}
+\subsubsection{Reproduding vs Classifing Quality}\label{sec:nogaequal}
When I started working on Anomaly Detection, I thougth that Autoencoder 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. This you can do 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, so after having a bit more experience encoding Data to classify it, and especcially when we now have another Method of seperating Data, I want to start this Chapter about why Autoencoder migth not be the best Idea, evaluating this hypotheosis.
To do this, lets look first at the loss of the Autoencoder: Since it is basically just the difference between Input and Output, it is a Measurement about how good the Autoencoder reproduces whatever we put into it\footnote{This is a bit of a simplification, since the Normation 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. Also, as argued for in the Chapter about Decicision Quality (ENTER CHAPTER LINK), we will use the AUC score to evalutate the Classifier Quality. So basically we want to see a strong falling correlation between the loss of my Network and the AUC score\footnote{The lower the loss, the higher the AUC}, do we see this? We show here the loss in training against the Decicion 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 cancelled by the Network not always reaching 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{W 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}.
@@ -1519,7 +1659,7 @@ As a final note in this subchapter: Please consider that oneoff networks should
(one could also note, that the same curve looks factually the same for a nongraph one, but less beautiful,but one could also not do this, since this is a point for graph oneoffs and I am not using graph oneoffs)
-\subsubsection{General Problems}\label{sec:General Problems}
+\subsubsection{General Problems}\label{sec:nogaegeneral}
These were quite specific problems to top tagging and anomaly detection, but there are some more general Problems working with Autoencoders, I want to focus here on two:
@@ -1527,7 +1667,7 @@ Firstly, it is not trivial to find the best compression size. The lower this siz
and also there is this image (circle reconstruction problem compared to oneoffs)
-\subsubsection{Autoencoder and losses}\label{sec:Autoencoder and losses}
+\subsubsection{Autoencoder and losses}\label{sec:nogaeloss}
talk about the mean guessing nets
@@ -1538,7 +1678,7 @@ talk about the mean guessing nets
%from file ..\..\write\/data\08anhang\01title
\newpage
-\section{Appendix}\label{sec:Appendix}
+\section{Appendix}\label{sec:appendix}