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2
latex/thesis/chapters/appendix.tex
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@ -104,7 +104,7 @@ The iterative algorithm can then be expressed as%
%
Using the definition of the proximal operator, the $\tilde{\boldsymbol{c}}$ update step
can be rewritten to match the definition given in section \ref{sec:dec:LP Decoding using ADMM}:%
can be rewritten to match the definition given in section \ref{sec:lp:Decoding Algorithm}:%
%
\begin{align*}
\tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}

62
latex/thesis/chapters/analysis_of_results.tex → latex/thesis/chapters/comparison.tex
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@ -1,42 +1,22 @@
\chapter{Analysis of Results}%
\label{chapter:Analysis of Results}
\chapter{Comparison of Proximal Decoding and \acs{LP} Decoding using \acs{ADMM}}%
\label{chapter:comparison}
TODO
%In this chapter, proximal decoding and \ac{LP} Decoding using \ac{ADMM} are compared.
%First the two algorithms are compared on a theoretical basis.
%Subsequently, their respective simulation results are examined and their
%differences interpreted on the basis of their theoretical structure.
%
%some similarities between the proximal decoding algorithm
%and \ac{LP} decoding using \ac{ADMM} are be pointed out.
%The two algorithms are compared and their different computational and decoding
%performance is interpreted on the basis of their theoretical structure.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{LP Decoding using ADMM}%
\label{sec:ana:LP Decoding using ADMM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proximal Decoding}%
\label{sec:ana:Proximal Decoding}
\begin{itemize}
\item Parameter choice
\item FER
\item Improved implementation
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Comparison of BP, Proximal Decoding and LP Decoding using ADMM}%
\label{sec:ana:Comparison of BP, Proximal Decoding and LP Decoding using ADMM}
\begin{itemize}
\item Decoding performance
\item Complexity \& runtime(mention difficulty in reaching conclusive
results when comparing implementations)
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theoretical Comparison of Proximal Decoding and LP Decoding using ADMM}%
\label{sec:Theoretical Comparison of Proximal Decoding and LP Decoding using ADMM}
In this section, some similarities between the proximal decoding algorithm
and \ac{LP} decoding using \ac{ADMM} are be pointed out.
The two algorithms are compared and their different computational and decoding
performance is interpreted on the basis of their theoretical structure.
\section{Theoretical Comparison}%
\label{sec:comp:theo}
\ac{ADMM} and the proximal gradient method can both be expressed in terms of
proximal operators.
@ -154,6 +134,8 @@ The advantage which arises because of this when using \ac{ADMM} is that
it can be easily detected, when the algorithm gets stuck - the algorithm
returns a pseudocodeword, the components of which are fractional.
\todo{Compare time complexity using Big-O notation}
\begin{itemize}
\item The comparison of actual implementations is always debatable /
contentious, since it is difficult to separate differences in
@ -169,3 +151,11 @@ returns a pseudocodeword, the components of which are fractional.
\item Proximal decoding faster than \ac{ADMM} $\rightarrow$ dafuq
(larger number of iterations before convergence?)
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Comparison of Results}%
\label{sec:comp:res}
TODO

440
latex/thesis/chapters/decoding_techniques.tex → latex/thesis/chapters/lp_dec_using_admm.tex
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@ -1,172 +1,12 @@
\chapter{Decoding Techniques}%
\label{chapter:decoding_techniques}
\chapter{\acs{LP} Decoding using \acs{ADMM}}%
\label{chapter:lp_dec_using_admm}
In this chapter, the decoding techniques examined in this work are detailed.
First, an overview of the general methodology of using optimization methods
for channel decoding is given.
Then, the field of \ac{LP} decoding and an \ac{ADMM}-based \ac{LP} decoding
algorithm are introduced.
Finally, the \textit{proximal decoding} algorithm is presented.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Decoding using Optimization Methods}%
\label{sec:dec:Decoding using Optimization Methods}
%
% General methodology
%
The general idea behind using optimization methods for channel decoding
is to reformulate the decoding problem as an optimization problem.
This new formulation can then be solved with one of the many
available optimization algorithms.
Generally, the original decoding problem considered is either the \ac{MAP} or
the \ac{ML} decoding problem:%
%
\begin{align}
\hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
p_{\boldsymbol{C} \mid \boldsymbol{Y}} \left(\boldsymbol{c} \mid \boldsymbol{y}
\right) \label{eq:dec:map}\\
\hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c}
\right) \label{eq:dec:ml}
.\end{align}%
%
The goal is to arrive at a formulation, where a certain objective function
$g : \mathbb{R}^n \rightarrow \mathbb{R} $ must be minimized under certain constraints:%
%
\begin{align*}
\text{minimize}\hspace{2mm} &g\left( \tilde{\boldsymbol{c}} \right)\\
\text{subject to}\hspace{2mm} &\tilde{\boldsymbol{c}} \in D
,\end{align*}%
%
where $D \subseteq \mathbb{R}^n$ is the domain of values attainable for $\tilde{\boldsymbol{c}}$
and represents the constraints.
In contrast to the established message-passing decoding algorithms,
the prespective then changes from observing the decoding process in its
Tanner graph representation with \acp{VN} and \acp{CN} (as shown in figure \ref{fig:dec:tanner})
to a spatial representation (figure \ref{fig:dec:spatial}),
where the codewords are some of the edges of a hypercube.
The goal is to find the point $\tilde{\boldsymbol{c}}$,
which minimizes the objective function $g$.
%
% Figure showing decoding space
%
\begin{figure}[H]
\centering
\begin{subfigure}[c]{0.47\textwidth}
\centering
\tikzstyle{checknode} = [color=KITblue, fill=KITblue,
draw, regular polygon,regular polygon sides=4,
inner sep=0pt, minimum size=12pt]
\tikzstyle{variablenode} = [color=KITgreen, fill=KITgreen,
draw, circle, inner sep=0pt, minimum size=10pt]
\begin{tikzpicture}[scale=1, transform shape]
\node[checknode,
label={[below, label distance=-0.4cm, align=center]
\acs{CN}\\$\left( c_1 + c_2 + c_3 = 0 \right) $}]
(cn) at (0, 0) {};
\node[variablenode, label={[above, align=center] \acs{VN}\\$\left( c_1 \right)$}]
(c1) at (-2, 2) {};
\node[variablenode, label={[above, align=center] \acs{VN}\\$\left( c_2 \right)$}]
(c2) at (0, 2) {};
\node[variablenode, label={[above, align=center] \acs{VN}\\$\left( c_3 \right)$}]
(c3) at (2, 2) {};
\draw (cn) -- (c1);
\draw (cn) -- (c2);
\draw (cn) -- (c3);
\end{tikzpicture}
\caption{Tanner graph representation of a single parity-check code}
\label{fig:dec:tanner}
\end{subfigure}%
\hfill%
\begin{subfigure}[c]{0.47\textwidth}
\centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
% Cube
\coordinate (p000) at (0, 0, 0);
\coordinate (p001) at (0, 0, 2);
\coordinate (p010) at (0, 2, 0);
\coordinate (p011) at (0, 2, 2);
\coordinate (p100) at (2, 0, 0);
\coordinate (p101) at (2, 0, 2);
\coordinate (p110) at (2, 2, 0);
\coordinate (p111) at (2, 2, 2);
\draw[] (p000) -- (p100);
\draw[] (p100) -- (p101);
\draw[] (p101) -- (p001);
\draw[] (p001) -- (p000);
\draw[dashed] (p010) -- (p110);
\draw[] (p110) -- (p111);
\draw[] (p111) -- (p011);
\draw[dashed] (p011) -- (p010);
\draw[dashed] (p000) -- (p010);
\draw[] (p100) -- (p110);
\draw[] (p101) -- (p111);
\draw[] (p001) -- (p011);
% Polytope Vertices
\node[codeword] (c000) at (p000) {};
\node[codeword] (c101) at (p101) {};
\node[codeword] (c110) at (p110) {};
\node[codeword] (c011) at (p011) {};
% Polytope Edges
% \draw[line width=1pt, color=KITblue] (c000) -- (c101);
% \draw[line width=1pt, color=KITblue] (c000) -- (c110);
% \draw[line width=1pt, color=KITblue] (c000) -- (c011);
%
% \draw[line width=1pt, color=KITblue] (c101) -- (c110);
% \draw[line width=1pt, color=KITblue] (c101) -- (c011);
%
% \draw[line width=1pt, color=KITblue] (c011) -- (c110);
% Polytope Annotations
\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, right=0.17cm of c101] {$\left( 1, 0, 1 \right) $};
\node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
% c
\node[color=KITgreen, fill=KITgreen,
draw, circle, inner sep=0pt, minimum size=4pt] (c) at (0.9, 0.7, 1) {};
\node[color=KITgreen, right=0cm of c] {$\tilde{\boldsymbol{c}}$};
\end{tikzpicture}
\caption{Spatial representation of a single parity-check code}
\label{fig:dec:spatial}
\end{subfigure}%
\caption{Different representations of the decoding problem}
\end{figure}
TODO
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{LP Decoding}%
\label{sec:dec:LP Decoding}
\label{sec:lp:LP Decoding}
\Ac{LP} decoding is a subject area introduced by Feldman et al.
\cite{feldman_paper}. They reframe the decoding problem as an
@ -276,7 +116,7 @@ the transfer matrix would be \cite[Sec. II, A]{efficient_lp_dec_admm}
.\end{align*}%
%
In figure \ref{fig:dec:poly}, the two relaxations are compared for an
In figure \ref{fig:lp:poly}, the two relaxations are compared for an
examplary code, which is described by the generator and parity-check matrices%
%
\begin{align}
@ -298,13 +138,13 @@ and has only two possible codewords:
\begin{bmatrix} 0 & 1 & 1 \end{bmatrix} \right\}
.\end{align*}
%
Figure \ref{fig:dec:poly:exact_ilp} shows the domain of exact \ac{ML} decoding.
Figure \ref{fig:lp:poly:exact_ilp} shows the domain of exact \ac{ML} decoding.
The first relaxation, onto the codeword polytope $\text{poly}\left( \mathcal{C} \right) $,
is shown in figure \ref{fig:dec:poly:exact};
is shown in figure \ref{fig:lp:poly:exact};
this expresses the constraints for the equivalent linear program to exact \ac{ML} decoding.
$\text{poly}\left( \mathcal{C} \right) $ is further relaxed onto the relaxed codeword polytope
$\overline{Q}$, shown in figure \ref{fig:dec:poly:relaxed}.
Figure \ref{fig:dec:poly:local} shows how $\overline{Q}$ is formed by intersecting the
$\overline{Q}$, shown in figure \ref{fig:lp:poly:relaxed}.
Figure \ref{fig:lp:poly:local} shows how $\overline{Q}$ is formed by intersecting the
local codeword polytopes of each check node.
%
%
@ -368,7 +208,7 @@ local codeword polytopes of each check node.
\end{tikzpicture}
\caption{Set of all codewords $\mathcal{C}$}
\label{fig:dec:poly:exact_ilp}
\label{fig:lp:poly:exact_ilp}
\end{subfigure}\\[1em]
\begin{subfigure}{\textwidth}
\centering
@ -429,7 +269,7 @@ local codeword polytopes of each check node.
\end{tikzpicture}
\caption{Codeword polytope $\text{poly}\left( \mathcal{C} \right) $}
\label{fig:dec:poly:exact}
\label{fig:lp:poly:exact}
\end{subfigure}
\end{subfigure} \hfill%
%
@ -574,7 +414,7 @@ local codeword polytopes of each check node.
\end{tikzpicture}
\caption{Local codeword polytopes of the check nodes}
\label{fig:dec:poly:local}
\label{fig:lp:poly:local}
\end{subfigure}\\[1em]
\begin{subfigure}{\textwidth}
\centering
@ -648,7 +488,7 @@ local codeword polytopes of each check node.
\end{tikzpicture}
\caption{Relaxed codeword polytope $\overline{Q}$}
\label{fig:dec:poly:relaxed}
\label{fig:lp:poly:relaxed}
\end{subfigure}
\end{subfigure}
@ -666,7 +506,7 @@ local codeword polytopes of each check node.
\caption{Visualization of the codeword polytope and the relaxed codeword
polytope of the code described by equations (\ref{eq:lp:example_code_def_gen})
and (\ref{eq:lp:example_code_def_par})}
\label{fig:dec:poly}
\label{fig:lp:poly}
\end{figure}%
%
\noindent It can be seen that the relaxed codeword polytope $\overline{Q}$ introduces
@ -689,10 +529,10 @@ The resulting formulation of the relaxed optimization problem becomes%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{LP Decoding using ADMM}%
\label{sec:dec:LP Decoding using ADMM}
\section{Decoding Algorithm}%
\label{sec:lp:Decoding Algorithm}
The \ac{LP} decoding formulation in section \ref{sec:dec:Decoding using Optimization Methods}
The \ac{LP} decoding formulation in section \ref{sec:lp:LP Decoding}
is a very general one that can be solved with a number of different optimization methods.
In this work \ac{ADMM} is examined, as its distributed nature allows for a very efficient
implementation.
@ -879,246 +719,12 @@ have been proposed (e.g., in \cite{original_admm}, \cite{efficient_lp_dec_admm},
The method chosen here is the one presented in \cite{lautern}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Implementation Details}%
\label{sec:lp:Implementation Details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proximal Decoding}%
\label{sec:dec:Proximal Decoding}
\section{Results}%
\label{sec:lp:Results}
Proximal decoding was proposed by Wadayama et. al as a novel formulation of
optimization-based decoding \cite{proximal_paper}.
With this algorithm, minimization is performed using the proximal gradient
method.
In contrast to \ac{LP} decoding, the objective function is based on a
non-convex optimization formulation of the \ac{MAP} decoding problem.
In order to derive the objective function, the authors begin with the
\ac{MAP} decoding rule, expressed as a continuous maximization problem%
\footnote{The expansion of the domain to be continuous doesn't constitute a
material difference in the meaning of the rule.
The only change is that what previously were \acp{PMF} now have to be expressed
in terms of \acp{PDF}.}
over $\boldsymbol{x}$:%
%
\begin{align}
\hat{\boldsymbol{x}} = \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}}
f_{\tilde{\boldsymbol{X}} \mid \boldsymbol{Y}}
\left( \tilde{\boldsymbol{x}} \mid \boldsymbol{y} \right)
= \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}} f_{\boldsymbol{Y}
\mid \tilde{\boldsymbol{X}}}
\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)%
\label{eq:prox:vanilla_MAP}
.\end{align}%
%
The likelihood $f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) $ is a known function
determined by the channel model.
The prior \ac{PDF} $f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$ is also
known, as the equal probability assumption is made on
$\mathcal{C}$.
However, since the considered domain is continuous,
the prior \ac{PDF} cannot be ignored as a constant during the minimization
as is often done, and has a rather unwieldy representation:%
%
\begin{align}
f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right) =
\frac{1}{\left| \mathcal{C} \right| }
\sum_{\boldsymbol{c} \in \mathcal{C} }
\delta\big( \tilde{\boldsymbol{x}} - \left( -1 \right) ^{\boldsymbol{c}}\big)
\label{eq:prox:prior_pdf}
.\end{align}%
%
In order to rewrite the prior \ac{PDF}
$f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$,
the so-called \textit{code-constraint polynomial} is introduced as:%
%
\begin{align*}
h\left( \tilde{\boldsymbol{x}} \right) =
\underbrace{\sum_{i=1}^{n} \left( \tilde{x_i}^2-1 \right) ^2}_{\text{Bipolar constraint}}
+ \underbrace{\sum_{j=1}^{m} \left[
\left( \prod_{i\in N_c \left( j \right) } \tilde{x_i} \right)
-1 \right] ^2}_{\text{Parity constraint}}%
.\end{align*}%
%
The intention of this function is to provide a way to penalize vectors far
from a codeword and favor those close to one.
In order to achieve this, the polynomial is composed of two parts: one term
representing the bipolar constraint, providing for a discrete solution of the
continuous optimization problem, and one term representing the parity
constraints, accommodating the role of the parity-check matrix $\boldsymbol{H}$.
The prior \ac{PDF} is then approximated using the code-constraint polynomial as:%
%
\begin{align}
f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)
\approx \frac{1}{Z}\mathrm{e}^{-\gamma h\left( \tilde{\boldsymbol{x}} \right) }%
\label{eq:prox:prior_pdf_approx}
.\end{align}%
%
The authors justify this approximation by arguing, that for
$\gamma \rightarrow \infty$, the approximation in equation
(\ref{eq:prox:prior_pdf_approx}) approaches the original function in equation
(\ref{eq:prox:prior_pdf}).
This approximation can then be plugged into equation (\ref{eq:prox:vanilla_MAP})
and the likelihood can be rewritten using the negative log-likelihood
$L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) = -\ln\left(
f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}\left(
\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) \right) $:%
%
\begin{align*}
\hat{\boldsymbol{x}} &= \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}}
\mathrm{e}^{- L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) }
\mathrm{e}^{-\gamma h\left( \tilde{\boldsymbol{x}} \right) } \\
&= \argmin_{\tilde{\boldsymbol{x}} \in \mathbb{R}^n} \big(
L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
+ \gamma h\left( \tilde{\boldsymbol{x}} \right)
\big)%
.\end{align*}%
%
Thus, with proximal decoding, the objective function
$g\left( \tilde{\boldsymbol{x}} \right)$ considered is%
%
\begin{align}
g\left( \tilde{\boldsymbol{x}} \right) = L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}}
\right)
+ \gamma h\left( \tilde{\boldsymbol{x}} \right)%
\label{eq:prox:objective_function}
\end{align}%
%
and the decoding problem is reformulated to%
%
\begin{align*}
\text{minimize}\hspace{2mm} &L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
+ \gamma h\left( \tilde{\boldsymbol{x}} \right)\\
\text{subject to}\hspace{2mm} &\tilde{\boldsymbol{x}} \in \mathbb{R}^n
.\end{align*}
%
For the solution of the approximate \ac{MAP} decoding problem, the two parts
of equation (\ref{eq:prox:objective_function}) are considered separately:
the minimization of the objective function occurs in an alternating
fashion, switching between the negative log-likelihood
$L\left( \boldsymbol{y} \mid \boldsymbol{x} \right) $ and the scaled
code-constraint polynomial $\gamma h\left( \boldsymbol{x} \right) $.
Two helper variables, $\boldsymbol{r}$ and $\boldsymbol{s}$, are introduced,
describing the result of each of the two steps.
The first step, minimizing the log-likelihood, is performed using gradient
descent:%
%
\begin{align}
\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \nabla
L\left( \boldsymbol{y} \mid \boldsymbol{s} \right),
\hspace{5mm}\omega > 0
\label{eq:prox:step_log_likelihood}
.\end{align}%
%
For the second step, minimizing the scaled code-constraint polynomial, the
proximal gradient method is used and the \textit{proximal operator} of
$\gamma h\left( \tilde{\boldsymbol{x}} \right) $ has to be computed.
It is then immediately approximated with gradient-descent:%
%
\begin{align*}
\textbf{prox}_{\gamma h} \left( \tilde{\boldsymbol{x}} \right) &\equiv
\argmin_{\boldsymbol{t} \in \mathbb{R}^n}
\left( \gamma h\left( \boldsymbol{t} \right) +
\frac{1}{2} \lVert \boldsymbol{t} - \tilde{\boldsymbol{x}} \rVert \right)\\
&\approx \tilde{\boldsymbol{x}} - \gamma \nabla h \left( \tilde{\boldsymbol{x}} \right),
\hspace{5mm} \gamma > 0, \text{ small}
.\end{align*}%
%
The second step thus becomes%
%
\begin{align*}
\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right),
\hspace{5mm}\gamma > 0,\text{ small}
.\end{align*}
%
While the approximation of the prior \ac{PDF} made in equation (\ref{eq:prox:prior_pdf_approx})
theoretically becomes better
with larger $\gamma$, the constraint that $\gamma$ be small is important,
as it keeps the effect of $h\left( \tilde{\boldsymbol{x}} \right) $ on the landscape
of the objective function small.
Otherwise, unwanted stationary points, including local minima, are introduced.
The authors say that ``in practice, the value of $\gamma$ should be adjusted
according to the decoding performance.'' \cite[Sec. 3.1]{proximal_paper}.
%The components of the gradient of the code-constraint polynomial can be computed as follows:%
%%
%\begin{align*}
% \frac{\partial}{\partial x_k} h\left( \boldsymbol{x} \right) =
% 4\left( x_k^2 - 1 \right) x_k + \frac{2}{x_k}
% \sum_{i\in \mathcal{B}\left( k \right) } \left(
% \left( \prod_{j\in\mathcal{A}\left( i \right)} x_j\right)^2
% - \prod_{j\in\mathcal{A}\left( i \right) }x_j \right)
%.\end{align*}%
%\todo{Only multiplication?}%
%\todo{$x_k$: $k$ or some other indexing variable?}%
%%
In the case of \ac{AWGN}, the likelihood
$f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)$
is%
%
\begin{align*}
f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
= \frac{1}{\sqrt{2\pi\sigma^2}}\mathrm{e}^{
-\frac{\lVert \boldsymbol{y}-\tilde{\boldsymbol{x}}
\rVert^2 }
{2\sigma^2}}
.\end{align*}
%
Thus, the gradient of the negative log-likelihood becomes%
\footnote{For the minimization, constants can be disregarded. For this reason,
it suffices to consider only proportionality instead of equality.}%
%
\begin{align*}
\nabla L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
&\propto -\nabla \lVert \boldsymbol{y} - \tilde{\boldsymbol{x}} \rVert^2\\
&\propto \tilde{\boldsymbol{x}} - \boldsymbol{y}
,\end{align*}%
%
allowing equation (\ref{eq:prox:step_log_likelihood}) to be rewritten as%
%
\begin{align*}
\boldsymbol{r} \leftarrow \boldsymbol{s}
- \omega \left( \boldsymbol{s} - \boldsymbol{y} \right)
.\end{align*}
%
One thing to consider during the actual decoding process, is that the gradient
of the code-constraint polynomial can take on extremely large values.
To avoid numerical instability, an additional step is added, where all
components of the current estimate are clipped to $\left[-\eta, \eta \right]$,
where $\eta$ is a positive constant slightly larger than one:%
%
\begin{align*}
\boldsymbol{s} \leftarrow \Pi_{\eta} \left( \boldsymbol{r}
- \gamma \nabla h\left( \boldsymbol{r} \right) \right)
,\end{align*}
%
$\Pi_{\eta}\left( \cdot \right) $ expressing the projection onto
$\left[ -\eta, \eta \right]^n$.
The iterative decoding process resulting from these considerations is shown in
figure \ref{fig:prox:alg}.
\begin{figure}[H]
\centering
\begin{genericAlgorithm}[caption={}, label={}]
$\boldsymbol{s} \leftarrow \boldsymbol{0}$
for $K$ iterations do
$\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \left( \boldsymbol{s} - \boldsymbol{y} \right) $
$\boldsymbol{s} \leftarrow \Pi_\eta \left(\boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) \right)$
$\boldsymbol{\hat{x}} \leftarrow \text{sign}\left( \boldsymbol{s} \right) $
if $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ do
return $\boldsymbol{\hat{c}}$
end if
end for
return $\boldsymbol{\hat{c}}$
\end{genericAlgorithm}
\caption{Proximal decoding algorithm for an \ac{AWGN} channel}
\label{fig:prox:alg}
\end{figure}

34
latex/thesis/chapters/methodology_and_implementation.tex
View File

@ -1,34 +0,0 @@
\chapter{Methodology and Implementation}%
\label{chapter:methodology_and_implementation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{General implementation process}%
\label{sec:impl:General implementation process}
\begin{itemize}
\item First python using numpy
\item Then C++ using Eigen
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{LP Decoding using ADMM}%
\label{sec:impl:LP Decoding using ADMM}
\begin{itemize}
\item Choice of parameters
\item Selected projection algorithm
\item Adaptive linear programming decoding?
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proximal Decoding}%
\label{sec:impl:Proximal Decoding}
\begin{itemize}
\item Choice of parameters
\item Road to improved implemenation
\end{itemize}

264
latex/thesis/chapters/proximal_decoding.tex Normal file
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@ -0,0 +1,264 @@
\chapter{Proximal Decoding}%
\label{chapter:proximal_decoding}
TODO
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Decoding Algorithm}%
\label{sec:prox:Decoding Algorithm}
Proximal decoding was proposed by Wadayama et. al as a novel formulation of
optimization-based decoding \cite{proximal_paper}.
With this algorithm, minimization is performed using the proximal gradient
method.
In contrast to \ac{LP} decoding, the objective function is based on a
non-convex optimization formulation of the \ac{MAP} decoding problem.
In order to derive the objective function, the authors begin with the
\ac{MAP} decoding rule, expressed as a continuous maximization problem%
\footnote{The expansion of the domain to be continuous doesn't constitute a
material difference in the meaning of the rule.
The only change is that what previously were \acp{PMF} now have to be expressed
in terms of \acp{PDF}.}
over $\boldsymbol{x}$:%
%
\begin{align}
\hat{\boldsymbol{x}} = \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}}
f_{\tilde{\boldsymbol{X}} \mid \boldsymbol{Y}}
\left( \tilde{\boldsymbol{x}} \mid \boldsymbol{y} \right)
= \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}} f_{\boldsymbol{Y}
\mid \tilde{\boldsymbol{X}}}
\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)%
\label{eq:prox:vanilla_MAP}
.\end{align}%
%
The likelihood $f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) $ is a known function
determined by the channel model.
The prior \ac{PDF} $f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$ is also
known, as the equal probability assumption is made on
$\mathcal{C}$.
However, since the considered domain is continuous,
the prior \ac{PDF} cannot be ignored as a constant during the minimization
as is often done, and has a rather unwieldy representation:%
%
\begin{align}
f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right) =
\frac{1}{\left| \mathcal{C} \right| }
\sum_{\boldsymbol{c} \in \mathcal{C} }
\delta\big( \tilde{\boldsymbol{x}} - \left( -1 \right) ^{\boldsymbol{c}}\big)
\label{eq:prox:prior_pdf}
.\end{align}%
%
In order to rewrite the prior \ac{PDF}
$f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$,
the so-called \textit{code-constraint polynomial} is introduced as:%
%
\begin{align*}
h\left( \tilde{\boldsymbol{x}} \right) =
\underbrace{\sum_{i=1}^{n} \left( \tilde{x_i}^2-1 \right) ^2}_{\text{Bipolar constraint}}
+ \underbrace{\sum_{j=1}^{m} \left[
\left( \prod_{i\in N_c \left( j \right) } \tilde{x_i} \right)
-1 \right] ^2}_{\text{Parity constraint}}%
.\end{align*}%
%
The intention of this function is to provide a way to penalize vectors far
from a codeword and favor those close to one.
In order to achieve this, the polynomial is composed of two parts: one term
representing the bipolar constraint, providing for a discrete solution of the
continuous optimization problem, and one term representing the parity
constraints, accommodating the role of the parity-check matrix $\boldsymbol{H}$.
The prior \ac{PDF} is then approximated using the code-constraint polynomial as:%
%
\begin{align}
f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)
\approx \frac{1}{Z}\mathrm{e}^{-\gamma h\left( \tilde{\boldsymbol{x}} \right) }%
\label{eq:prox:prior_pdf_approx}
.\end{align}%
%
The authors justify this approximation by arguing, that for
$\gamma \rightarrow \infty$, the approximation in equation
(\ref{eq:prox:prior_pdf_approx}) approaches the original function in equation
(\ref{eq:prox:prior_pdf}).
This approximation can then be plugged into equation (\ref{eq:prox:vanilla_MAP})
and the likelihood can be rewritten using the negative log-likelihood
$L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) = -\ln\left(
f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}\left(
\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) \right) $:%
%
\begin{align*}
\hat{\boldsymbol{x}} &= \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}}
\mathrm{e}^{- L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) }
\mathrm{e}^{-\gamma h\left( \tilde{\boldsymbol{x}} \right) } \\
&= \argmin_{\tilde{\boldsymbol{x}} \in \mathbb{R}^n} \big(
L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
+ \gamma h\left( \tilde{\boldsymbol{x}} \right)
\big)%
.\end{align*}%
%
Thus, with proximal decoding, the objective function
$g\left( \tilde{\boldsymbol{x}} \right)$ considered is%
%
\begin{align}
g\left( \tilde{\boldsymbol{x}} \right) = L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}}
\right)
+ \gamma h\left( \tilde{\boldsymbol{x}} \right)%
\label{eq:prox:objective_function}
\end{align}%
%
and the decoding problem is reformulated to%
%
\begin{align*}
\text{minimize}\hspace{2mm} &L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
+ \gamma h\left( \tilde{\boldsymbol{x}} \right)\\
\text{subject to}\hspace{2mm} &\tilde{\boldsymbol{x}} \in \mathbb{R}^n
.\end{align*}
%
For the solution of the approximate \ac{MAP} decoding problem, the two parts
of equation (\ref{eq:prox:objective_function}) are considered separately:
the minimization of the objective function occurs in an alternating
fashion, switching between the negative log-likelihood
$L\left( \boldsymbol{y} \mid \boldsymbol{x} \right) $ and the scaled
code-constraint polynomial $\gamma h\left( \boldsymbol{x} \right) $.
Two helper variables, $\boldsymbol{r}$ and $\boldsymbol{s}$, are introduced,
describing the result of each of the two steps.
The first step, minimizing the log-likelihood, is performed using gradient
descent:%
%
\begin{align}
\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \nabla
L\left( \boldsymbol{y} \mid \boldsymbol{s} \right),
\hspace{5mm}\omega > 0
\label{eq:prox:step_log_likelihood}
.\end{align}%
%
For the second step, minimizing the scaled code-constraint polynomial, the
proximal gradient method is used and the \textit{proximal operator} of
$\gamma h\left( \tilde{\boldsymbol{x}} \right) $ has to be computed.
It is then immediately approximated with gradient-descent:%
%
\begin{align*}
\textbf{prox}_{\gamma h} \left( \tilde{\boldsymbol{x}} \right) &\equiv
\argmin_{\boldsymbol{t} \in \mathbb{R}^n}
\left( \gamma h\left( \boldsymbol{t} \right) +
\frac{1}{2} \lVert \boldsymbol{t} - \tilde{\boldsymbol{x}} \rVert \right)\\
&\approx \tilde{\boldsymbol{x}} - \gamma \nabla h \left( \tilde{\boldsymbol{x}} \right),
\hspace{5mm} \gamma > 0, \text{ small}
.\end{align*}%
%
The second step thus becomes%
%
\begin{align*}
\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right),
\hspace{5mm}\gamma > 0,\text{ small}
.\end{align*}
%
While the approximation of the prior \ac{PDF} made in equation (\ref{eq:prox:prior_pdf_approx})
theoretically becomes better
with larger $\gamma$, the constraint that $\gamma$ be small is important,
as it keeps the effect of $h\left( \tilde{\boldsymbol{x}} \right) $ on the landscape
of the objective function small.
Otherwise, unwanted stationary points, including local minima, are introduced.
The authors say that ``in practice, the value of $\gamma$ should be adjusted
according to the decoding performance.'' \cite[Sec. 3.1]{proximal_paper}.
%The components of the gradient of the code-constraint polynomial can be computed as follows:%
%%
%\begin{align*}
% \frac{\partial}{\partial x_k} h\left( \boldsymbol{x} \right) =
% 4\left( x_k^2 - 1 \right) x_k + \frac{2}{x_k}
% \sum_{i\in \mathcal{B}\left( k \right) } \left(
% \left( \prod_{j\in\mathcal{A}\left( i \right)} x_j\right)^2
% - \prod_{j\in\mathcal{A}\left( i \right) }x_j \right)
%.\end{align*}%
%\todo{Only multiplication?}%
%\todo{$x_k$: $k$ or some other indexing variable?}%
%%
In the case of \ac{AWGN}, the likelihood
$f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)$
is%
%
\begin{align*}
f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
= \frac{1}{\sqrt{2\pi\sigma^2}}\mathrm{e}^{
-\frac{\lVert \boldsymbol{y}-\tilde{\boldsymbol{x}}
\rVert^2 }
{2\sigma^2}}
.\end{align*}
%
Thus, the gradient of the negative log-likelihood becomes%
\footnote{For the minimization, constants can be disregarded. For this reason,
it suffices to consider only proportionality instead of equality.}%
%
\begin{align*}
\nabla L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
&\propto -\nabla \lVert \boldsymbol{y} - \tilde{\boldsymbol{x}} \rVert^2\\
&\propto \tilde{\boldsymbol{x}} - \boldsymbol{y}
,\end{align*}%
%
allowing equation (\ref{eq:prox:step_log_likelihood}) to be rewritten as%
%
\begin{align*}
\boldsymbol{r} \leftarrow \boldsymbol{s}
- \omega \left( \boldsymbol{s} - \boldsymbol{y} \right)
.\end{align*}
%
One thing to consider during the actual decoding process, is that the gradient
of the code-constraint polynomial can take on extremely large values.
To avoid numerical instability, an additional step is added, where all
components of the current estimate are clipped to $\left[-\eta, \eta \right]$,
where $\eta$ is a positive constant slightly larger than one:%
%
\begin{align*}
\boldsymbol{s} \leftarrow \Pi_{\eta} \left( \boldsymbol{r}
- \gamma \nabla h\left( \boldsymbol{r} \right) \right)
,\end{align*}
%
$\Pi_{\eta}\left( \cdot \right) $ expressing the projection onto
$\left[ -\eta, \eta \right]^n$.
The iterative decoding process resulting from these considerations is shown in
figure \ref{fig:prox:alg}.
\begin{figure}[H]
\centering
\begin{genericAlgorithm}[caption={}, label={}]
$\boldsymbol{s} \leftarrow \boldsymbol{0}$
for $K$ iterations do
$\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \left( \boldsymbol{s} - \boldsymbol{y} \right) $
$\boldsymbol{s} \leftarrow \Pi_\eta \left(\boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) \right)$
$\boldsymbol{\hat{x}} \leftarrow \text{sign}\left( \boldsymbol{s} \right) $
if $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ do
return $\boldsymbol{\hat{c}}$
end if
end for
return $\boldsymbol{\hat{c}}$
\end{genericAlgorithm}
\caption{Proximal decoding algorithm for an \ac{AWGN} channel}
\label{fig:prox:alg}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Implementation Details}%
\label{sec:prox:Implementation Details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}%
\label{sec:prox:Results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Improved Implementation}%
\label{sec:prox:Improved Implementation}

161
latex/thesis/chapters/theoretical_background.tex
View File

@ -5,10 +5,11 @@ In this chapter, the theoretical background necessary to understand this
work is given.
First, the used notation is clarified.
The physical aspects are detailed - the used modulation scheme and channel model.
A short introduction of channel coding with binary linear codes and especially
A short introduction to channel coding with binary linear codes and especially
\ac{LDPC} codes is given.
The established methods of decoding LPDC codes are briefly explained.
Lastly, the optimization methods utilized are described.
Lastly, the general process of decoding using optimization techniques is described
and an overview of the utilized optimization methods is given.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -270,6 +271,161 @@ the use of \ac{BP} impractical for applications where a very low \ac{BER} is
desired \cite[Sec. 15.3]{ryan_lin_2009}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Decoding using Optimization Methods}%
\label{sec:theo:Decoding using Optimization Methods}
%
% General methodology
%
The general idea behind using optimization methods for channel decoding
is to reformulate the decoding problem as an optimization problem.
This new formulation can then be solved with one of the many
available optimization algorithms.
Generally, the original decoding problem considered is either the \ac{MAP} or
the \ac{ML} decoding problem:%
%
\begin{align}
\hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
p_{\boldsymbol{C} \mid \boldsymbol{Y}} \left(\boldsymbol{c} \mid \boldsymbol{y}
\right) \label{eq:dec:map}\\
\hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c}
\right) \label{eq:dec:ml}
.\end{align}%
%
The goal is to arrive at a formulation, where a certain objective function
$g : \mathbb{R}^n \rightarrow \mathbb{R} $ must be minimized under certain constraints:%
%
\begin{align*}
\text{minimize}\hspace{2mm} &g\left( \tilde{\boldsymbol{c}} \right)\\
\text{subject to}\hspace{2mm} &\tilde{\boldsymbol{c}} \in D
,\end{align*}%
%
where $D \subseteq \mathbb{R}^n$ is the domain of values attainable for $\tilde{\boldsymbol{c}}$
and represents the constraints.
In contrast to the established message-passing decoding algorithms,
the prespective then changes from observing the decoding process in its
Tanner graph representation with \acp{VN} and \acp{CN} (as shown in figure \ref{fig:dec:tanner})
to a spatial representation (figure \ref{fig:dec:spatial}),
where the codewords are some of the edges of a hypercube.
The goal is to find the point $\tilde{\boldsymbol{c}}$,
which minimizes the objective function $g$.
%
% Figure showing decoding space
%
\begin{figure}[H]
\centering
\begin{subfigure}[c]{0.47\textwidth}
\centering
\tikzstyle{checknode} = [color=KITblue, fill=KITblue,
draw, regular polygon,regular polygon sides=4,
inner sep=0pt, minimum size=12pt]
\tikzstyle{variablenode} = [color=KITgreen, fill=KITgreen,
draw, circle, inner sep=0pt, minimum size=10pt]
\begin{tikzpicture}[scale=1, transform shape]
\node[checknode,
label={[below, label distance=-0.4cm, align=center]
\acs{CN}\\$\left( c_1 + c_2 + c_3 = 0 \right) $}]
(cn) at (0, 0) {};
\node[variablenode, label={[above, align=center] \acs{VN}\\$\left( c_1 \right)$}]
(c1) at (-2, 2) {};
\node[variablenode, label={[above, align=center] \acs{VN}\\$\left( c_2 \right)$}]
(c2) at (0, 2) {};
\node[variablenode, label={[above, align=center] \acs{VN}\\$\left( c_3 \right)$}]
(c3) at (2, 2) {};
\draw (cn) -- (c1);
\draw (cn) -- (c2);
\draw (cn) -- (c3);
\end{tikzpicture}
\caption{Tanner graph representation of a single parity-check code}
\label{fig:dec:tanner}
\end{subfigure}%
\hfill%
\begin{subfigure}[c]{0.47\textwidth}
\centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
% Cube
\coordinate (p000) at (0, 0, 0);
\coordinate (p001) at (0, 0, 2);
\coordinate (p010) at (0, 2, 0);
\coordinate (p011) at (0, 2, 2);
\coordinate (p100) at (2, 0, 0);
\coordinate (p101) at (2, 0, 2);
\coordinate (p110) at (2, 2, 0);
\coordinate (p111) at (2, 2, 2);
\draw[] (p000) -- (p100);
\draw[] (p100) -- (p101);
\draw[] (p101) -- (p001);
\draw[] (p001) -- (p000);
\draw[dashed] (p010) -- (p110);
\draw[] (p110) -- (p111);
\draw[] (p111) -- (p011);
\draw[dashed] (p011) -- (p010);
\draw[dashed] (p000) -- (p010);
\draw[] (p100) -- (p110);
\draw[] (p101) -- (p111);
\draw[] (p001) -- (p011);
% Polytope Vertices
\node[codeword] (c000) at (p000) {};
\node[codeword] (c101) at (p101) {};
\node[codeword] (c110) at (p110) {};
\node[codeword] (c011) at (p011) {};
% Polytope Edges
% \draw[line width=1pt, color=KITblue] (c000) -- (c101);
% \draw[line width=1pt, color=KITblue] (c000) -- (c110);
% \draw[line width=1pt, color=KITblue] (c000) -- (c011);
%
% \draw[line width=1pt, color=KITblue] (c101) -- (c110);
% \draw[line width=1pt, color=KITblue] (c101) -- (c011);
%
% \draw[line width=1pt, color=KITblue] (c011) -- (c110);
% Polytope Annotations
\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, right=0.17cm of c101] {$\left( 1, 0, 1 \right) $};
\node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
% c
\node[color=KITgreen, fill=KITgreen,
draw, circle, inner sep=0pt, minimum size=4pt] (c) at (0.9, 0.7, 1) {};
\node[color=KITgreen, right=0cm of c] {$\tilde{\boldsymbol{c}}$};
\end{tikzpicture}
\caption{Spatial representation of a single parity-check code}
\label{fig:dec:spatial}
\end{subfigure}%
\caption{Different representations of the decoding problem}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Optimization Methods}
\label{sec:theo:Optimization Methods}
@ -484,3 +640,4 @@ condition that the step size be $\mu$:%
% \hspace{5mm} \mu > 0
.\end{align*}
%

6
latex/thesis/thesis.tex
View File

@ -213,9 +213,9 @@
\include{chapters/introduction}
\include{chapters/theoretical_background}
\include{chapters/decoding_techniques}
\include{chapters/methodology_and_implementation}
\include{chapters/analysis_of_results}
\include{chapters/proximal_decoding}
\include{chapters/lp_dec_using_admm}
\include{chapters/comparison}
\include{chapters/discussion}
\include{chapters/conclusion}
\include{chapters/appendix}