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Added general methodology for optimization decoding; minor changes to proximal decoding

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Andreas Tsouchlos 2023-02-15 17:33:45 +01:00
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181
latex/thesis/chapters/decoding_techniques.tex
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@ -1,18 +1,50 @@
\chapter{Decoding Techniques}%
\label{chapter:decoding_techniques}
In this chapter, the decoding techniques examined in this work are detailed.
First, an overview of of the general methodology of using optimization methods
for channel decoding is given. Afterwards, the specific decoding techniques
themselves are explained.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Decoding using Optimization Methods}%
\label{sec:dec:Decoding using Optimization Methods}
%
% TODOs
% General methodology
%
\begin{itemize}
\item General methodology
\end{itemize}
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_{c \in \mathcal{C}}
f_{\boldsymbol{X} \mid \boldsymbol{Y}} \left( \boldsymbol{x} \mid \boldsymbol{y} \right)\\
\hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{c \in \mathcal{C}}
f_{\boldsymbol{Y} \mid \boldsymbol{X}} \left( \boldsymbol{y} \mid \boldsymbol{x} \right)
.\end{align*}
%
The goal is to arrive at a formulation, where a certain objective function
$f$ has to be minimized under certain constraints:%
%
\begin{align*}
\text{minimize } f\left( \boldsymbol{x} \right)\\
\text{subject to \ldots}
.\end{align*}
In contrast to the established message-passing decoding algorithms,
the viewpoint then changes from observing the decoding process in its
tanner graph based representation (as shown in figure \ref{fig:dec:tanner})
into a spacial representation, where the codewords are some of the edges
of a hypercube and the goal is to find that point $\boldsymbol{x}$,
\todo{$\boldsymbol{x}$? Or some other variable?}
which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spacial}).
%
% Figure showing decoding space
@ -21,48 +53,80 @@
\begin{figure}[H]
\centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\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]
\tdplotsetmaincoords{60}{245}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
% Cube
\begin{tikzpicture}[scale=1, transform shape]
\node[checknode,
label={[below, label distance=-0.4cm, align=center]
$c$\\$\left( x_1 + x_2 + x_3 = 0 \right) $}]
(c) at (0, 0) {};
\node[variablenode, label={$x_1$}] (x1) at (-2, 2) {};
\node[variablenode, label={$x_2$}] (x2) at (0, 2) {};
\node[variablenode, label={$x_3$}] (x3) at (2, 2) {};
\draw[dashed] (0, 0, 0) -- (2, 0, 0);
\draw[dashed] (2, 0, 0) -- (2, 0, 2);
\draw[] (2, 0, 2) -- (0, 0, 2);
\draw[] (0, 0, 2) -- (0, 0, 0);
\draw (c) -- (x1);
\draw (c) -- (x2);
\draw (c) -- (x3);
\end{tikzpicture}
\caption{Tanner graph representation of a single parity-check code}
\label{fig:dec:tanner}
\end{subfigure}%
\hfill%
\begin{subfigure}[c]{0.47\textwidth}
\centering
\draw[] (0, 2, 0) -- (2, 2, 0);
\draw[] (2, 2, 0) -- (2, 2, 2);
\draw[] (2, 2, 2) -- (0, 2, 2);
\draw[] (0, 2, 2) -- (0, 2, 0);
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\draw[] (0, 0, 0) -- (0, 2, 0);
\draw[dashed] (2, 0, 0) -- (2, 2, 0);
\draw[] (2, 0, 2) -- (2, 2, 2);
\draw[] (0, 0, 2) -- (0, 2, 2);
\tdplotsetmaincoords{60}{245}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
% Cube
% Polytope Annotations
\draw[dashed] (0, 0, 0) -- (2, 0, 0);
\draw[dashed] (2, 0, 0) -- (2, 0, 2);
\draw[] (2, 0, 2) -- (0, 0, 2);
\draw[] (0, 0, 2) -- (0, 0, 0);
\node[codeword] (c000) at (0, 0, 0) {};% {$\left( 0, 0, 0 \right) $};
\node[codeword] (c101) at (2, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
\node[codeword] (c110) at (2, 2, 0) {};% {$\left( 1, 1, 0 \right) $};
\node[codeword] (c011) at (0, 2, 2) {};% {$\left( 0, 1, 1 \right) $};
\draw[] (0, 2, 0) -- (2, 2, 0);
\draw[] (2, 2, 0) -- (2, 2, 2);
\draw[] (2, 2, 2) -- (0, 2, 2);
\draw[] (0, 2, 2) -- (0, 2, 0);
\node[color=KITblue, right=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c101] {$\left( 1, 0, 1 \right) $};
\node[color=KITblue, left=0cm of c110] {$\left( 1, 1, 0 \right) $};
\node[color=KITblue, left=-0.1cm of c011] {$\left( 0, 1, 1 \right) $};
\draw[] (0, 0, 0) -- (0, 2, 0);
\draw[dashed] (2, 0, 0) -- (2, 2, 0);
\draw[] (2, 0, 2) -- (2, 2, 2);
\draw[] (0, 0, 2) -- (0, 2, 2);
% f
% Polytope Annotations
\node[color=KITgreen, fill=KITgreen,
draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.9, 0.7, 1) {};
\node[color=KITgreen, right=0cm of f] {$\boldsymbol{f}$};
\end{tikzpicture}
\node[codeword] (c000) at (0, 0, 0) {};% {$\left( 0, 0, 0 \right) $};
\node[codeword] (c101) at (2, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
\node[codeword] (c110) at (2, 2, 0) {};% {$\left( 1, 1, 0 \right) $};
\node[codeword] (c011) at (0, 2, 2) {};% {$\left( 0, 1, 1 \right) $};
\caption{Hypercube ($n=3$) and valid codewords for a single parity-check code}
\node[color=KITblue, right=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c101] {$\left( 1, 0, 1 \right) $};
\node[color=KITblue, left=0cm of c110] {$\left( 1, 1, 0 \right) $};
\node[color=KITblue, left=-0.1cm of c011] {$\left( 0, 1, 1 \right) $};
% x
\node[color=KITgreen, fill=KITgreen,
draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.9, 0.7, 1) {};
\node[color=KITgreen, right=0cm of f] {$\boldsymbol{x}$};
\end{tikzpicture}
\caption{Spacial representation of a single parity-check code}
\label{fig:dec:spacial}
\end{subfigure}%
\end{figure}
@ -120,6 +184,7 @@ representing the bibolar constraint, providing for a discrete solution of the
continuous optimization problem, and one term representing the parity
constraint, accomodating the role of the parity-check matrix $\boldsymbol{H}$.
%
The equal probability assumption is made on $\mathcal{C}\left( \boldsymbol{H} \right) $.
The prior \ac{PDF} is then approximated using the code-constraint polynomial\todo{Italic?}:%
%
\begin{align}
@ -186,14 +251,14 @@ immediately approximalted by a gradient-descent step:%
\argmin_{\boldsymbol{t} \in \mathbb{R}^n}
\left( \gamma h\left( \boldsymbol{x} \right) +
\frac{1}{2} \lVert \boldsymbol{t} - \boldsymbol{x} \rVert \right)\\
&\approx \boldsymbol{x} - \gamma h \left( \boldsymbol{x} \right),
&\approx \boldsymbol{x} - \gamma \nabla h \left( \boldsymbol{r} \right),
\hspace{5mm} \gamma \text{ small}
.\end{align*}%
%
The second step thus becomes%
%
\begin{align*}
\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma h\left( \boldsymbol{x} \right),
\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right),
\hspace{5mm}\gamma > 0,\text{ small}
.\end{align*}
%
@ -205,8 +270,7 @@ 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.
The iterative decoding process resulting from this considreation is shown in
The iterative decoding process \todo{projection with $\eta$} resulting from this considreation is shown in
figure \ref{fig:prox:alg}.
\begin{figure}[H]
@ -230,3 +294,40 @@ return $\boldsymbol{\hat{c}}$
\label{fig:prox:alg}
\end{figure}
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?}%
%
In the case of \ac{AWGN}, the likelihood
$f_{\boldsymbol{Y} \mid \boldsymbol{X}}\left( \boldsymbol{y} \mid \boldsymbol{x} \right)$
is%
%
\begin{align*}
f_{\boldsymbol{Y} \mid \boldsymbol{X}}\left( \boldsymbol{y} \mid \boldsymbol{x} \right)
= \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{\lVert \boldsymbol{y}-\boldsymbol{x} \rVert^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 the proportionality instead of the equality}%
%
\begin{align*}
\nabla L \left( \boldsymbol{y} \mid \boldsymbol{x} \right)
&\propto -\nabla \lVert \boldsymbol{y} - \boldsymbol{x} \rVert^2\\
&\propto \boldsymbol{x} - \boldsymbol{y}
.\end{align*}%
%
The resulting iterative decoding process under the assumption of \ac{AWGN} is
described by%
%
\begin{align*}
\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega\left( \boldsymbol{s}-\boldsymbol{y} \right)\\
\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right)
.\end{align*}

1
latex/thesis/thesis.tex
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@ -33,6 +33,7 @@
\usetikzlibrary{external}
\usetikzlibrary{spy}
\usetikzlibrary{shapes.geometric}
\pgfplotsset{compat=newest}
\usepgfplotslibrary{colorbrewer}