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Added visualization of codeword polytope and relaxed codeword polytope

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Andreas Tsouchlos 2023-02-18 13:13:27 +01:00
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latex/thesis/chapters/decoding_techniques.tex
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@ -143,11 +143,13 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
\Ac{LP} decoding is a subject area introduced by Feldman et al.
\todo{Space before citation?}
\cite{feldman_paper}. They reframed the decoding problem as an
\textit{integer linear program} and subsequently presented a relaxation into
a \textit{linear program}, lifting the integer requirement.
The optimization method used to solve this problem that is examined in this
work is \ac{ADMM}.
\cite{feldman_paper}. They reframe the decoding problem as an
\textit{integer linear program} and subsequently present two relaxations into
\textit{linear programs}, one representing a formulation of exact \ac{LP}
decoding and an approximation with a more manageable representation.
To solve the resulting linear program, various optimization methods can be
used.
The one examined in this work is \ac{ADMM}.
\todo{Why chose ADMM?}
Feldman at al. begin by looking at the \ac{ML} decoding problem%
@ -189,73 +191,315 @@ decoding is the following:%
.\end{align*}%
%
As solving integer linear programs is generally NP-hard, the decoding problem \todo{New \S?}
As solving integer linear programs is generally NP-hard, this decoding problem
has to be approximated by one with looser constraints.
A technique called \textit{\ac{LP} Relaxation} is applied,
essentially removing the requirement for the components of $\boldsymbol{c}$
to be integer values.
In order to provide a formal definition of the relaxed constraints, the
authors go on to define the concept of the \textit{codeword polytope}
(figure \ref{fig:dec:poly}) as
being the convex hull of all possible codewords:
A technique called \textit{relaxation} is applied,
essentially modifying the constraints in order to broaden the considered
domain (e.g. by lifting the integer requirement).
First, the authors present an equivalent \ac{LP} formulation of exact \ac{ML}
decoding, redefining the constraints in terms of the \text{codeword polytope}
%
\begin{align*}
\text{poly}\left( \mathcal{C} \right) = \left\{
\sum_{c \in \mathcal{C}} \lambda_{\boldsymbol{c}} \boldsymbol{c}
\text{ : } \lambda_{\boldsymbol{c}} \ge 0,
\sum_{\boldsymbol{c} \in \mathcal{C}} \lambda_{\boldsymbol{c}} = 1 \right\}
.\end{align*}
,\end{align*} %
%
which represents the \textit{convex hull} of all possible codewords,
i.e. the set of convex linear combinations of all codewords
(visualized in figure \ref{fig:dec:poly}).
However, since the number of constraints needed to characterize this codeword
polytope is exponential in the code length, this formulation is relaxed futher.
%
% Codeword polytope visualization figure
%
\begin{figure}[H]
\centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
%
% Left side - codeword polytope
%
\tdplotsetmaincoords{60}{245}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
% Cube
\begin{subfigure}[c]{0.45\textwidth}
\centering
\begin{subfigure}{\textwidth}
\centering
\begin{align*}
\boldsymbol{H} &=
\begin{bmatrix}
1 & 1 & 1\\
0 & 1 & 1
\end{bmatrix}\\[1em]
\mathcal{C} &= \left\{
\begin{bmatrix}
0\\
0\\
0
\end{bmatrix},
\begin{bmatrix}
0\\
1\\
1
\end{bmatrix}
\right\}
\end{align*}
\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);
\caption{Definition of the visualized code}
\label{fig:}
\end{subfigure} \\[7em]
\begin{subfigure}{\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}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
% Cube
% Codeword Polytope
\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[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 0, 2);
\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 2, 0);
\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (0, 2, 2);
\draw[] (p000) -- (p100);
\draw[] (p100) -- (p101);
\draw[] (p101) -- (p001);
\draw[] (p001) -- (p000);
\draw[line width=1pt, color=KITblue] (2, 0, 2) -- (2, 2, 0);
\draw[line width=1pt, color=KITblue] (2, 0, 2) -- (0, 2, 2);
\draw[dashed] (p010) -- (p110);
\draw[] (p110) -- (p111);
\draw[] (p111) -- (p011);
\draw[dashed] (p011) -- (p010);
\draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);
\draw[dashed] (p000) -- (p010);
\draw[] (p100) -- (p110);
\draw[] (p101) -- (p111);
\draw[] (p001) -- (p011);
% Polytope Annotations
% Polytope Vertices
\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) $};
\node[codeword] (c000) at (p000) {};
\node[codeword] (c011) at (p011) {};
% Polytope Edges
\node[color=KITblue, right=0cm of c000] {$\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) $};
\end{tikzpicture}
\draw[line width=1pt, color=KITblue] (c000) -- (c011);
% Polytope Annotations
\caption{Codeword polytope of a single parity-check code}
\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
\end{tikzpicture}
\caption{Codeword polytope}
\label{fig:}
\end{subfigure}
\end{subfigure} \hfill%
%
%
% Right side - relaxed polytope
%
%
\begin{subfigure}[c]{0.45\textwidth}
\centering
\begin{subfigure}{\textwidth}
\centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
% Cube
\coordinate (p000) at (0, 0, 0);
\coordinate (p001) at (0, 0, 2);
\coordinate (p010) at (0, 2, 0);
\coordinate (p011) at (0, 2, 2);
\coordinate (p100) at (2, 0, 0);
\coordinate (p101) at (2, 0, 2);
\coordinate (p110) at (2, 2, 0);
\coordinate (p111) at (2, 2, 2);
\draw[] (p000) -- (p100);
\draw[] (p100) -- (p101);
\draw[] (p101) -- (p001);
\draw[] (p001) -- (p000);
\draw[dashed] (p010) -- (p110);
\draw[] (p110) -- (p111);
\draw[] (p111) -- (p011);
\draw[dashed] (p011) -- (p010);
\draw[dashed] (p000) -- (p010);
\draw[] (p100) -- (p110);
\draw[] (p101) -- (p111);
\draw[] (p001) -- (p011);
% Polytope Vertices
\node[codeword] (c000) at (p000) {};
\node[codeword] (c101) at (p101) {};
\node[codeword] (c110) at (p110) {};
\node[codeword] (c011) at (p011) {};
% Polytope Edges
\draw[line width=1pt, color=KITblue] (c000) -- (c101);
\draw[line width=1pt, color=KITblue] (c000) -- (c110);
\draw[line width=1pt, color=KITblue] (c000) -- (c011);
\draw[line width=1pt, color=KITblue] (c101) -- (c110);
\draw[line width=1pt, color=KITblue] (c101) -- (c011);
\draw[line width=1pt, color=KITblue] (c011) -- (c110);
% Polytope Annotations
\node[color=KITblue, below=0cm of c000] {$\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) $};
\end{tikzpicture}
\caption{Local codeword polytope of parity-check
$\begin{bmatrix} 1 & 1 & 1 \end{bmatrix}$}
\label{fig:}
\end{subfigure} \\[1em]
\begin{subfigure}{\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] (c011) at (p011) {};
\node[codeword] (c100) at (p100) {};
\node[codeword] (c111) at (p111) {};
% Polytope Edges
\draw[line width=1pt, color=KITblue] (c000) -- (c011);
\draw[line width=1pt, color=KITblue] (c000) -- (c100);
\draw[line width=1pt, color=KITblue] (c100) -- (c111);
\draw[line width=1pt, color=KITblue] (c111) -- (c011);
% Polytope Annotations
\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
\node[color=KITblue, below=0cm of c100] {$\left( 1, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c111] {$\left( 1, 1, 1 \right) $};
\end{tikzpicture}
\caption{Local codeword polytope of parity-check
$\begin{bmatrix} 0 & 1 & 1 \end{bmatrix}$}
\end{subfigure}\\[1em]
\begin{subfigure}{\textwidth}
\centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tikzstyle{pseudocodeword} = [color=KITred, fill=KITred,
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] (c011) at (p011) {};
\node[pseudocodeword] (cpseudo) at (2, 1, 1) {};
% Polytope Edges
\draw[line width=1pt, color=KITblue] (c000) -- (c011);
\draw[line width=1pt, color=KITred] (cpseudo) -- (c000);
\draw[line width=1pt, color=KITred] (cpseudo) -- (c011);
% Polytope Annotations
\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
\node[color=KITred, right=0.03cm of cpseudo]
{$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
\end{tikzpicture}
\caption{Relaxed codeword polytope}
\label{fig:}
\end{subfigure}
\end{subfigure}
\caption{Visualization of the codeword polytope and the relaxed codeword
polytope for an example code}
\label{fig:dec:poly}
\end{figure}