masterthesis-latex/body.tex

% body thesis file that contains the actual content

\chapter{Introduction}

\subsection*{Motivation}

Famous examples like the automatic soap dispenser which does not
recognize the hand of a black person but dispenses soap when presented
with a paper towel raise the question of bias in computer
systems~\cite{Friedman1996}. Related to this ethical question regarding
the design of so called algorithms is the question of
algorithmic accountability~\cite{Diakopoulos2014}.

Supervised neural networks learn from input-output relations and
figure out by themselves what connections are necessary for that.
This feature is also their Achilles heel: it makes them effectively
black boxes and prevents any answers to questions of causality.

However, these questions of causility are of enormous consequence when
results of neural networks are used to make life changing decisions:
Is a correlation enough to bring forth negative consequences
for a particular person? And if so, what is the possible defence
against math? Similar questions can be raised when looking at computer
vision networks that might be used together with so called smart
CCTV cameras to discover suspicious activity.

This leads to the need for neural networks to explain their results.
Such an explanation must come from the network or an attached piece
of technology to allow adoption in mass. Obviously this setting
poses the question, how such an endeavour can be achieved.

For neural networks there are fundamentally two type of tasks:
regression and classification. Regression deals with any case
where the goal for the network is to come close to an ideal
function that connects all data points. Classification, however,
describes tasks where the network is supposed to identify the
class of any given input. In this thesis, I will focus on
classification.

\subsection*{Object Detection in Open Set Conditions}

More specifically, I will look at object detection in the open set
conditions. In non-technical words this effectively describes
the kind of situation you encounter with CCTV cameras or robots
outside of a laboratory. Both use cameras that record
images. Subsequently a neural network analyses the image
and returns a list of detected and classified objects that it
found in the image. The problem here is that networks can only
classify what they know. If presented with an object type that
the network was not trained with, as happens frequently in real
environments, it will still classify the object and might even
have a high confidence in doing so. Such an example would be
a false positive. Any ordinary person who uses the results of
such a network would falsely assume that a high confidence always
means the classification is very likely correct. If they use
a proprietary system they might not even be able to find out
that the network was never trained on a particular type of object.
Therefore it would be impossible for them to identify the output
of the network as false positive.

This goes back to the need for automatic explanation. Such a system
should by itself recognize that the given object is unknown and
hence mark any classification result of the network as meaningless.
Technically there are two slightly different things that deal
with this type of task: model uncertainty and novelty detection.

Model uncertainty can be measured with dropout sampling.
Dropout is usually used only during training but
Miller et al.~\cite{Miller2018} use them also during testing
to achieve different results for the same image making use of
multiple forward passes. The output scores for the forward passes
of the same image are then averaged. If the averaged class
probabilities resemble a uniform distribution (every class has
the same probability) this symbolises maximum uncertainty. Conversely,
if there is one very high probability with every other being very
low this signifies a low uncertainty. An unknown object is more
likely to cause high uncertainty which allows for an identification
of false positive cases.

Novelty detection is the more direct approach to solve the task.
In the realm of neural networks it is usually done with the help of
auto-encoders that essentially solve a regression task of finding an
identity function that reconstructs on the output the given
input~\cite{Pimentel2014}. Auto-encoders have
internally at least two components: an encoder, and a decoder or
generator. The job of the encoder is to find an encoding that
compresses the input as good as possible while simultaneously
being as loss-free as possible. The decoder takes this latent
representation of the input and has to find a decompression
that reconstructs the input as accurate as possible. During
training these auto-encoders learn to reproduce a certain group
of object classes. The actual novelty detection takes place
during testing: Given an image, and the output and loss of the
auto-encoder, a novelty score is calculated. A low novelty
score signals a known object. The opposite is true for a high
novelty score.

\subsection*{Research Question}

Both presented approaches describe one way to solve the aforementioned
problem of explanation. They can be differentiated by measuring
their performance: the best theoretical idea is useless if it does
not perform well. Miller et al. have shown
some success in using dropout sampling. However, the many forward
passes during testing for every image seem computationally expensive.
In comparison a single run through a trained auto-encoder seems
intuitively to be faster. This leads to the hypothesis (see below).

For the purpose of this thesis, I will
use the work of Miller et al. as baseline to compare against.
They use the SSD~\cite{Liu2016} network for object detection,
modified by added dropout layers, and the SceneNet
RGB-D~\cite{McCormac2017} data set using the MS COCO~\cite{Lin2014}
classes. I will use a simple implementation of an auto-encoder and
novelty detection to compare with the work of Miller et al.
SSD for the object detection and SceneNet RGB-D as the data
set are used for both approaches.

\paragraph{Hypothesis} Novelty detection using auto-encoders
delivers similar or better object detection performance under open set
conditions while being less computationally expensive compared to
dropout sampling.

\paragraph{Contribution}
The contribution of this thesis is a comparison between dropout
sampling and auto-encoding with respect to the overall performance
of both for object detection in the open set conditions using
the SSD network for object detection and the SceneNet RGB-D data set
with MS COCO classes.

\chapter{Background and Contribution}

This chapter will begin with an overview over previous works
in the field of this thesis. Afterwards the theoretical foundations
of the work of Miller et al.~\cite{Miller2018} and auto-encoders will
be explained. The chapter concludes with more details about the
research question and the intended contribution of this thesis.

For both background sections the notation defined in table
\ref{tab:notation} will be used.

\section{Related Works}

Novelty detection for object detection is intricately linked with
open set conditions: the test data can contain unknown classes.
Bishop~\cite{Bishop1994} investigates the correlation between
the degree of novel input data and the reliability of network
outputs. Pimentel et al.~\cite{Pimentel2014} provide a review
of novelty detection methods published over the previous decade.

There are two primary pathways that deal with novelty: novelty
detection using auto-encoders and uncertainty estimation with
bayesian networks.

Japkowicz et al.~\cite{Japkowicz1995} introduce a novelty detection
method based on the hippocampus of Gluck and Meyers~\cite{Gluck1993}
and use an auto-encoder to recognize novel instances.
Thompson et al.~\cite{Thompson2002} show that auto-encoders
can learn "normal" system behaviour implicitly.
Goodfellow et al.~\cite{Goodfellow2014} introduce adversarial
networks: a generator that attempts to trick the discriminator
by generating samples indistinguishable from the real data.
Makhzani et al.~\cite{Makhzani2015} build on the work of Goodfellow
and propose adversarial auto-encoders. Richter and
Roy~\cite{Richter2017} use an auto-encoder to detect novelty.

Wang et al.~\cite{Wang2018} base upon Goodfellow's work and
use a generative adversarial network for novelty detection.
Sabokrou et al.~\cite{Sabokrou2018} implement an end-to-end
architecture for one-class classification: it consists of two
deep networks, with one being the novelty detector and the other
enhancing inliers and distorting outliers.
Pidhorskyi et al.~\cite{Pidhorskyi2018} take a probabilistic approach
and compute how likely it is that a sample is generated by the
inlier distribution.

Kendall and Gal~\cite{Kendall2017} provide a Bayesian deep learning
framework that combines input-dependent
aleatoric\footnote{captures noise inherent in observations}
uncertainty with epistemic\footnote{uncertainty in the model}
uncertainty. Lakshminarayanan et al.~\cite{Lakshminarayanan2017}
implement a predictive uncertainty estimation using deep ensembles
rather than Bayesian networks. Geifman et al.~\cite{Geifman2018}
introduce an uncertainty estimation algorithm for non-Bayesian deep
neural classification that estimates the uncertainty of highly
confident points using earlier snapshots of the trained model.
Miller et al.~\cite{Miller2018a} compare merging strategies
for sampling-based uncertainty techniques in object detection.
Sensoy et al.~\cite{Sensoy2018} treat prediction confidence
as subjective opinions: they place a Dirichlet distribution on it.
The trained predictor for a multi-class classification is also a
Dirichlet distribution.

Gal and Ghahramani~\cite{Gal2016} show how dropout can be used
as a Bayesian approximation. Miller et al.~\cite{Miller2018}
build upon the work of Miller et al.~\cite{Miller2018a} and
Gal and Ghahramani: they use dropout sampling under open-set
conditions for object detection. Mukhoti and Gal~\cite{Mukhoti2018}
contribute metrics to measure uncertainty for semantic
segmentation. Wu et al.~\cite{Wu2019} introduce two innovations
that turn variational Bayes into a robust tool for Bayesian
networks: they introduce a novel deterministic method to approximate
moments in neural networks which eliminates gradient variance, and
they introduce a hierarchical prior for parameters and an
Empirical Bayes procedure to select prior variances.

\section{Background for Bayesian SSD}

\begin{table}
    \centering
    \caption{Notation for background sections}
    \label{tab:notation}
    \begin{tabular}{l|l}
        symbol & meaning \\
        \hline
        \(\mathbf{W}\) & weights \\
        \(\mathbf{T}\) & training data \\
        \(\mathcal{N}(0, I)\) & Gaussian distribution \\
        \(I\) & independent and identical distribution \\
        \(p(\mathbf{W}|\mathbf{T})\) & probability of weights given
            training data \\
        \(\mathcal{I}\) & an image \\
        \(\mathbf{q} = p(y|\mathcal{I}, \mathbf{T})\) & probability
            of all classes given image and training data \\
        \(H(\mathbf{q})\) & entropy over probability vector \\
        \(\widetilde{\mathbf{W}}\) & weights sampled from
            \(p(\mathbf{W}|\mathbf{T})\) \\
        \(\mathbf{b}\) & bounding box coordinates \\
        \(\mathbf{s}\) & softmax scores \\
        \(\overline{\mathbf{s}}\) & averaged softmax scores \\
        \(D\) & detections of one forward pass \\
        \(\mathfrak{D}\) & set of all detections over multiple
            forward passes \\
        \(\mathcal{O}\) & observation \\
        \(\overline{\mathbf{q}}\) & probability vector for
            observation \\
        \(\overline{\mathbf{z}}, \mathbf{z}\) & latent space representation \\
        \(d_T, d_z\) & discriminators \\
        \(e, g\) & encoding and decoding/generating function \\
        \(J_g\) & Jacobi matrix for generating function \\
        \(\mathcal{T}\) & tangent space \\
        \(\mathbf{R}\) & training/test data changed to be on tangent space
    \end{tabular}
\end{table}

To understand dropout sampling, it is necessary to explain the
idea of Bayesian neural networks. They place a prior distribution
over the network weights, for example a Gaussian prior distribution:
\(\mathbf{W} \sim \mathcal{N}(0, I)\). In this example
\(\mathbf{W}\) are the weights and \(I\) symbolises that every
weight is drawn from an independent and identical distribution. The
training of the network determines a plausible set of weights by
evaluating the posterior (probability output) over the weights given
the training data: \(p(\mathbf{W}|\mathbf{T})\). However, this
evaluation cannot be performed in any reasonable
time. Therefore approximation techniques are
required. In those techniques the posterior is fitted with a
simple distribution \(q^{*}_{\theta}(\mathbf{W})\). The original
and intractable problem of averaging over all weights in the network
is replaced with an optimisation task, where the parameters of the
simple distribution are optimised over~\cite{Kendall2017}.

\subsubsection*{Dropout Variational Inference}

Kendall and Gal~\cite{Kendall2017} showed an approximation for
classfication and recognition tasks. Dropout variational inference
is a practical approximation technique by adding dropout layers
in front of every weight layer and using them also during test
time to sample from the approximate posterior. Effectively, this
results in the approximation of the class probability
\(p(y|\mathcal{I}, \mathbf{T})\) by performing multiple forward
passes through the network and averaging over the obtained Softmax
scores \(\mathbf{s}_i\), given an image \(\mathcal{I}\) and the
training data \(\mathbf{T}\):
\begin{equation} \label{eq:drop-sampling}
p(y|\mathcal{I}, \mathbf{T}) = \int p(y|\mathcal{I}, \mathbf{W}) \cdot p(\mathbf{W}|\mathbf{T})d\mathbf{W} \approx \frac{1}{n} \sum_{i=1}^{n}\mathbf{s}_i
\end{equation}

With this dropout sampling technique \(n\) model weights
\(\widetilde{\mathbf{W}}_i\) are sampled from the posterior
\(p(\mathbf{W}|\mathbf{T})\). The class probability
\(p(y|\mathcal{I}, \mathbf{T})\) is a probability vector
\(\mathbf{q}\) over all class labels. Finally, the uncertainty
of the network with respect to the classification is given by
the entropy \(H(\mathbf{q}) = - \sum_i q_i \cdot \log q_i\).

\subsubsection*{Dropout Sampling for Object Detection}

Miller et al.~\cite{Miller2018} apply the dropout sampling to
object detection. In that case \(\mathbf{W}\) represents the
learned weights of a detection network like SSD~\cite{Liu2016}.
Every forward pass uses a different network
\(\widetilde{\mathbf{W}}\) which is approximately sampled from
\(p(\mathbf{W}|\mathbf{T})\). Each forward pass in object
detection results in a set of detections, each consisting of bounding
box coordinates \(\mathbf{b}\) and softmax score \(\mathbf{s}\).
The detections are denoted by Miller et al. as \(D_i =
\{\mathbf{s}_i,\mathbf{b}_i\}\). The detections of all passes are put
into a large set \(\mathfrak{D} = \{D_1, ..., D_2\}\).

All detections with mutual intersection-over-union scores (IoU)
of \(0.95\) or higher are defined as an observation \(\mathcal{O}_i\).
Subsequently, the corresponding vector of class probabilities
\(\overline{\mathbf{q}}_i\) for the observation is calculated by averaging all
score vectors \(\mathbf{s}_j\) in a particular observation
\(\mathcal{O}_i\): \(\overline{\mathbf{q}}_i \approx \overline{\mathbf{s}}_i = \frac{1}{n} \sum_{j=1}^{n} \mathbf{s}_j\). The label uncertainty
of the detector for a particular observation is measured by
the entropy \(H(\overline{\mathbf{q}}_i) = - \sum_j \overline{q}_{ij} \cdot \log \overline{q}_{ij}\).

In the introduction I used a very reduced version to describe
maximum and low uncertainty. A more complete explanation:
If \(\overline{\mathbf{q}}_i\), which I called averaged class probabilities,
resembles a uniform distribution the entropy will be high. A uniform
distribution means that no class is more likely than another, which
is a perfect example of maximum uncertainty. Conversely, if
one class has a very high probability the entropy will be low.

In open set conditions it can be expected that falsely generated
detections for unknown object classes have a higher label
uncertainty. A treshold on the entropy \(H(\overline{\mathbf{q}}_i)\) can then
be used to identify and reject these false positive cases.

\section{Adversarial Auto-encoder}

This section will explain the adversarial auto-encoder used by
Pidhorskyi et al.~\cite{Pidhorskyi2018} but in a slightly modified
form to make it more understandable.

The training data \(\mathbf{T} \in \mathbb{R}^m \) is the input
of the auto-encoder. An encoding function \(e: \mathbb{R}^m \rightarrow \mathbb{R}^n\) takes the data
and produces a representation \(\overline{\mathbf{z}} \in \mathbb{R}^n\)
in a latent space. This latent space is smaller (\(n < m\)) than the
input which necessitates some form of compression.

A second function \(g: \Omega \rightarrow \mathbb{R}^m\) is the
generator function that takes the latent representation
\(\mathbf{z} \in \Omega \subset \mathbb{R}^n\) and generates an output
\(\overline{\mathbf{T}}\) as close as possible to the input data
distribution.

What then is the difference between \(\overline{\mathbf{z}}\) and \(\mathbf{z}\)?
With a simple auto-encoder both would be identical. In this case
of an adversarial auto-encoder it is slightly more complicated.
There is a discriminator \(d_z\) that tries to distinguish between
an encoded data point \(\overline{\mathbf{z}}\) and a \(\mathbf{z} \sim \mathcal{N}(0,1)\) drawn from a normal distribution with \(0\) mean
and a standard deviation of \(1\). During training, the encoding
function \(e\) attempts to minimize any perceivable difference
between \(\mathbf{z}\) and \(\overline{\mathbf{z}}\) while \(d_z\) has the
aforementioned adversarial task to differentiate between them.

Furthermore, there is a discriminator \(d_T\) that has the task
to differentiate the generated output \(\overline{\mathbf{T}}\) from the
actual input \(\mathbf{T}\). During training, the generator function \(g\)
tries to minimize the perceivable difference between \(\overline{\mathbf{T}}\) and \(\mathbf{T}\) while \(d_T\) has the mentioned
adversarial task to distinguish between them.

With this all components of the adversarial auto-encoder employed
by Pidhorskyi et al are introduced. Finally, the losses are
presented. The two adversarial objectives have been mentioned
already. Specifically, there is the adversarial loss for the
discriminator \(d_z\):
\begin{equation} \label{eq:adv-loss-z}
    \mathcal{L}_{adv-d_z}(\mathbf{T},e,d_z) = E[\log (d_z(\mathcal{N}(0,1)))] + E[\log (1 - d_z(e(\mathbf{T})))],
\end{equation}
\noindent
where \(E\) stands for an expected
value\footnote{a term used in probability theory},
\(\mathbf{T}\) stands for the input, and
\(\mathcal{N}(0,1)\) represents an element drawn from the specified
distribution. The encoder \(e\) attempts to minimize this loss while
the discriminator \(d_z\) intends to maximize it.

In the same way the adversarial loss for the discriminator \(d_T\)
is specified:
\begin{equation} \label{eq:adv-loss-x}
    \mathcal{L}_{adv-d_T}(\mathbf{T},d_T,g) = E[\log(d_T(\mathbf{T}))] + E[\log(1 - d_T(g(\mathcal{N}(0,1))))],
\end{equation}
\noindent
where \(\mathbf{T}\), \(E\), and \(\mathcal{N}(0,1)\) have the same meaning
as before. In this case the generator \(g\) tries to minimize the loss
while the discriminator \(d_T\) attempts to maximize it.

Every auto-encoder requires a reconstruction error to work. This
error calculates the difference between the original input and
the generated or decoded output. In this case, the reconstruction
loss is defined like this:
\begin{equation} \label{eq:recon-loss}
    \mathcal{L}_{error}(\mathbf{T}, e, g) = - E[\log(p(g(e(\mathbf{T})) | \mathbf{T}))],
\end{equation}
\noindent
where \(\log(p)\) is the expected log-likelihood and \(\mathbf{T}\),
\(E\), \(e\), and \(g\) have the same meaning as before.

All losses combined result in the following formula:
\begin{equation} \label{eq:full-loss}
    \mathcal{L}(\mathbf{T},e,d_z,d_T,g) = \mathcal{L}_{adv-d_z}(\mathbf{T},e,d_z) + \mathcal{L}_{adv-d_T}(x,d_T,g) + \lambda \mathcal{L}_{error}(\mathbf{T},e,g),
\end{equation}
\noindent
where \(\lambda\) is a parameter used to balance the adversarial
losses with the reconstruction loss. The model is trained by
Pidhorskyi et al using the Adam optimizer by doing alternative
updates of each of the aforementioned components:

\begin{itemize}
    \item Maximize \(\mathcal{L}_{adv-d_T}\) by updating weights of \(d_T\);
    \item Minimize \(\mathcal{L}_{adv-d_T}\) by updating weights of \(g\);
    \item Maximize \(\mathcal{L}_{adv-d_z}\) by updating weights of \(d_z\);
    \item Minimize \(\mathcal{L}_{error}\) and \(\mathcal{L}_{adv-d_z}\) by updating weights of \(e\) and \(g\).
\end{itemize}

Practically, the auto-encoder is trained separately for every
object class that is considered "known". Pidhorskyi et al trained
it on the MNIST~\cite{Lecun1998} data set, once for every digit.

For this thesis it needs to be trained on the SceneNet RGB-D
data set using MS COCO classes as known classes. As in every
test epoch all known classes are present, it becomes
non-trivial which of the trained auto-encoders should be used to
calculate novelty. To phrase it differently, a true positive
detection is possible for multiple classes in the same image.
If, for example, one object is classified correctly by SSD as a chair
the novelty score should be low. But the auto-encoders of all
known classes but the "chair" class will give ideally a high novelty
score. Which of the values should be used? The only sensible solution
is to only run it through the auto-encoder that was trained for
the class the SSD model predicted. This provides the following
scenarios:
\begin{itemize}
    \item true positive classification: novelty score should be low
    \item false positive classification and correct class is
    among the known classes: novelty score should be high
    \item false positive classification and correct class is unknown:
    novelty score should be high
\end{itemize}
\noindent
Negative classifications are not listed as these are not part
of the output of the SSD and cannot be given to the auto-encoder
as input. Furthermore, the 2nd case should not happen because
the trained SSD knows this other class and is very likely
to give it a higher probability. Therefore, using only one
auto-encoder fulfils the task of differentiating between
known and unknown classes.

\section{Generative Probabilistic Novelty Detection}

It is still unclear how the novelty score is calculated.
This section will clear this up in as understandable as
possible terms. However, the name "Generative Probabilistic
Novelty Detection"~\cite{Pidhorskyi2018} already signals that
probability theory has something to do with it. Furthermore, this
section will make use of some mathematical terms which cannot
be explained in great detail here. Moreover, the previous section
already introduced many required components, which will not be
explained here again.

For the purpose of this explanation a trained auto-encoder
is assumed. In that case the generator function describes
the model that the auto-encoder is actually using for the
novelty detection. The task of training is to make sure this
model comes as close as possible to the real model of the
training or testing data. The model of the auto-encoder
is in mathematical terms a parameterized manifold
\(\mathcal{M} \equiv g(\Omega)\) of dimension \(n\).
The set of training or testing data can then be described
in the following way:
\begin{equation} \label{eq:train-set}
    \mathbf{T} = g(\mathbf{z}) + \xi_i \quad i \in \mathbb{N},
\end{equation}
\noindent
where \(\xi_i\) represents noise. It may be confusing but
for the purpose of this novelty test the "truth" is what
the generator function generates from a set of \(\mathbf{z} \in \Omega\),
not the ground truth from the data set. Furthermore,
the previously introduced encoder function \(e\) is assumed
to work as an exact inverse of \(g\) for every \(\mathbf{T} \in \mathcal{M}\).
For such \(\mathbf{T}\) it follows that \(\mathbf{T} = g(e(\mathbf{T}))\).

Let \(\overline{\mathbf{T}} \in \mathbb{R}^m\) be the test data. The
remainder of the section will explain how the novelty
test is performed for this \(\overline{\mathbf{T}}\). It is important
to note that this data is not necessarily part of the
auto-encoder model. Therefore, \(g(e(\overline{\mathbf{T}})) = \mathbf{T}\) cannot
be assumed. However, it can be observed that \(\overline{\mathbf{T}}\)
can be non-linearly projected onto
\(\overline{\mathbf{T}}^{\|} \in \mathcal{M}\)
by using \(g(\overline{\mathbf{z}})\) with \(\overline{\mathbf{z}} = e(\overline{\mathbf{T}})\).
It is assumed that \(g\) is smooth enough to perform a linearization
based on the first-order Taylor expansion:
\begin{equation} \label{eq:taylor-expanse}
    g(\mathbf{z}) = g(\overline{\mathbf{z}}) + J_g(\overline{\mathbf{z}}) (\mathbf{z} - \overline{\mathbf{z}}) + \mathcal{O}(\| \mathbf{z} - \overline{\mathbf{z}} \|^2),
\end{equation}
\noindent
where \(J_g(\overline{\mathbf{z}})\) is the Jacobi matrix of \(g\) computed
at \(\overline{\mathbf{z}}\). It is assumed that the Jacobi matrix of \(g\)
has the full rank at every point of the manifold. A Jacobi matrix
contains all first-order partial derivatives of a function.
\(\| \cdot \|\) is the \(\mathbf{L}_2\) norm, which calculates the
length of a vector by calculating the square root of the sum of
squares of all dimensions of the vector. Lastly, \(\mathcal{O}\)
is called Big-O notation and is used for specifying the time
complexity of an algorithm. In this case it contains a linear
value, which means that this part of the term can be ignored for
\(\mathbf{z}\) growing to infinity.

Next the tangent space of \(g\) at \(\overline{\mathbf{T}}^{\|}\), which
is spanned by the \(n\) independent column vectors of the Jacobi
matrix \(J_g(\overline{\mathbf{z}})\), is defined as
\(\mathcal{T} = \text{span}(J_g(\overline{\mathbf{z}}))\). The tangent space
of a point of a function describes all the vectors that could go
through this point. The Jacobi matrix can be decomposed into three
matrices using singular value decomposition: \(J_g(\overline{\mathbf{z}}) = U^{\|}SV^{*}\). \(\mathcal{T}\) is defined to also be spanned
by the column vectors of \(U^{\|}\): \(\mathcal{T} = \text{span}(U^{\|})\). \(U^{\|}\) contains the left-singular values
and \(V^{*}\) is the conjugate transposed version of the matrix
\(V\), which contains the right-singular values. \(U^{\bot}\) is
defined in such a way that \(U = [U^{\|}U^{\bot}]\) is a unitary
matrix. \(\mathcal{T^{\bot}}\) is the orthogonal complement of
\(\mathcal{T}\). With this preparation \(\overline{\mathbf{T}}\) can be
represented with respect to the local coordinates that define
\(\mathcal{T}\) and \(\mathcal{T}^{\bot}\). This representation
can be achieved by computing
\begin{equation} \label{eq:w-definition}
    \overline{\mathbf{R}} = U^{\top} \overline{\mathbf{T}} = \left[\begin{matrix}
        U^{\|^{\top}} \overline{\mathbf{T}} \\
        U^{\bot^{\top}} \overline{\mathbf{T}}
    \end{matrix}\right] = \left[\begin{matrix}
        \overline{\mathbf{R}}^{\|} \\
        \overline{\mathbf{R}}^{\bot}
    \end{matrix}\right],
\end{equation}
\noindent
where the rotated coordinates (training/testing data points
changed to be on the tangent space)
\(\overline{\mathbf{R}}\) are decomposed into \(\overline{\mathbf{R}}^{\|}\), which
are parallel to \(\mathcal{T}\), and \(\overline{\mathbf{R}}^{\bot}\), which
are orthogonal to \(\mathcal{T}\).

The last step to define the novelty test involves probability
density functions (PDFs), which are now introduced. The PDF \(p_T(\mathbf{T})\)
describes the random variable \(T\), from which the training and
testing data points are drawn. In addition, \(p_R(\mathbf{R})\) is the
probability density function of the random variable \(W\),
which represents \(T\) after changing the coordinates. Both
distributions are identical. But it is assumed that the coordinates
\(R^{\|}\), which are parallel to \(\mathcal{T}\), and the coordinates
\(R^{\bot}\), which are orthogonal to \(\mathcal{T}\), are
statistically independent. With this assumption the following holds:
\begin{equation} \label{eq:pdf-x}
    p_T(\mathbf{T}) = p_R(\mathbf{R}) = p_R(\mathbf{R}^{\|}, \mathbf{R}^{\bot}) = p_{R^{\|}}(\mathbf{R}^{\|}) p_{R^{\bot}}(\mathbf{R}^{\bot})
\end{equation}
The previously introduced noise comes into play again. In formula
(\ref{eq:train-set}) it is assumed that the noise \(\xi\)
predominantly deviates the data points \(\mathbf{T}\) away from the manifold
\(\mathcal{M}\) in a direction orthogonal to \(\mathcal{T}\).
As a consequence \(R^{\bot}\) is mainly responsible for the noise
effects. Since noise and drawing from the manifold are statistically
independent, \(R^{\|}\) and \(R^{\bot}\) are also independent.

Finally, referring back to the data point \(\overline{\mathbf{T}}\), the
novelty test is defined like this:
\begin{equation} \label{eq:novelty-test}
    p_T(\overline{\mathbf{T}}) = p_{R^{\|}}(\overline{\mathbf{R}}^{\|})p_{R^{\bot}}(\overline{\mathbf{R}}^{\bot}) =
    \begin{cases}
        \geq \gamma & \Longrightarrow \text{Inlier} \\
        < \gamma &  \Longrightarrow \text{Outlier}
    \end{cases}
\end{equation}
\noindent
where \(\gamma\) is a suitable threshold.

At this point it is very clear that the GPND approach requires
far more math background than dropout sampling to understand
the novelty test. Nonetheless it could be the better method.

% SSD: \cite{Liu2016}
% ImageNet: \cite{Deng2009}
% COCO: \cite{Lin2014}
% YCB: \cite{Xiang2017}
% SceneNet: \cite{McCormac2017}

\chapter{Methods}

This chapter starts with the design of the source code; the
source code is so much more than a means to an end. The thesis
uses two data sets: MS COCO and SceneNet RGB-D; a section
will explain how these data sets have been prepared.
Afterwards the replication of the work of Miller et al. is
outlined, followed by the implementation of the auto-encoder.

\section{Design of Source Code}

The source code of many published papers is either not available
or seems like an afterthought: it is poorly documented, difficult
to integrate in your own work, and often does not follow common
software development best practices. Moreover, with Tensorflow,
PyTorch, and Caffe there are at least three machine learning
frameworks. Every research team seems to prefer another framework
and sometimes even develops their own; this makes it difficult
to combine the work of different authors.
In addition to all this, most papers do not contain proper information
regarding the implementation details, making it difficult to
accurately replicate them if their source code is not available.

Therefore, it was clear to me: I will release my source code and
make it available as Python package on the PyPi package index.
This makes it possible for other researchers to simply install
a package and use the API to interact with my code. Additionally,
the code has been designed to be future proof and work with
the announced Tensorflow 2.0 by supporting eager mode.

Furthermore, it is configurable, well documented, and conforms
to the clean code guidelines: evolvability and extendability among
others. Unit tests are part of the code as well to identify common
issues early on, saving time in the process.

Lastly, the SSD implementation from a third party repository
has been modified to work inside a Python package architecture and
with eager mode.

\section{Preparation of data sets}

Usually, data sets are not perfect when it comes to neural
networks: they contain outliers, invalid bounding boxes, and similar
problematic things. Before a data set can be used, these problems
need to be removed.

For the MS COCO data set, all annotations were checked for
impossible values: bounding box height or width lower than zero,
x1 and y1 bounding box coordinates lower than zero,
x2 and y2 coordinates lower or equal to zero, x1 greater than x2,
y1 greater than y2, image width lower than x2,
and image height lower than y2. In the last two cases the
bounding box width or height was set to (image with - x1) or (image height - y1)
respectively; in the other cases the annotation was skipped.
If the bounding box width or height afterwards is
lower or equal to zero the annotation is skipped.

In this thesis SceneNet RGB-D is always used with COCO classes.
Therefore, a mapping between COCO and SceneNet RGB-D and vice versa
was necessary. It was created my manually going through each
Wordnet ID and searching for a fitting COCO class.

The ground truth for SceneNet RGB-D is stored in protobuf files
and had to be converted into Python format to use it in the
codebase. Only ground truth instances that had a matching
COCO class were saved, the rest discarded. 

\section{Replication of Miller et al.}

\section{Implementing an auto-encoder}

\chapter{Results}

\chapter{Discussion}

\chapter{Closing}