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Modified figure to be clearer; Moved footnote into text

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Andreas Tsouchlos 2023-03-08 10:54:49 +01:00
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latex/thesis/chapters/decoding_techniques.tex
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@ -27,7 +27,7 @@ the \ac{ML} decoding problem:%
% %
\begin{align*} \begin{align*}
\hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}} \hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
P \left(\boldsymbol{C} = \boldsymbol{c} \mid \boldsymbol{Y} = \boldsymbol{y} P \left(\boldsymbol{c} \mid \boldsymbol{Y} = \boldsymbol{y}
\right)\\ \right)\\
\hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}} \hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c} f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c}
@ -35,7 +35,7 @@ the \ac{ML} decoding problem:%
.\end{align*}% .\end{align*}%
% %
The goal is to arrive at a formulation, where a certain objective function The goal is to arrive at a formulation, where a certain objective function
$g \left( \cdot \right) $ must be minimized under certain constraints:% $g : \mathbb{R}^n \rightarrow \mathbb{R}^n $ must be minimized under certain constraints:%
% %
\begin{align*} \begin{align*}
\text{minimize}\hspace{2mm} &g\left( \tilde{\boldsymbol{c}} \right)\\ \text{minimize}\hspace{2mm} &g\left( \tilde{\boldsymbol{c}} \right)\\
@ -51,7 +51,7 @@ Tanner graph representation with \acp{VN} and \acp{CN} (as shown in figure \ref{
to a spatial representation (figure \ref{fig:dec:spatial}), to a spatial representation (figure \ref{fig:dec:spatial}),
where the codewords are some of the edges of a hypercube. where the codewords are some of the edges of a hypercube.
The goal is to find the point $\tilde{\boldsymbol{c}}$, The goal is to find the point $\tilde{\boldsymbol{c}}$,
which minimizes the objective function $g\left( \cdot \right) $. which minimizes the objective function $g$.
% %
% Figure showing decoding space % Figure showing decoding space
@ -162,8 +162,6 @@ which minimizes the objective function $g\left( \cdot \right) $.
\caption{Different representations of the decoding problem} \caption{Different representations of the decoding problem}
\end{figure} \end{figure}
\todo{Rename $c$ to e.g. $h$ or remove it completely?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -222,7 +220,7 @@ decoding is the following:%
%Especially for the continuous variable in LP decoding} %Especially for the continuous variable in LP decoding}
As solving integer linear programs is generally NP-hard, this decoding problem As solving integer linear programs is generally NP-hard, this decoding problem
has to be approximated by one with looser constraints. has to be approximated by a problem with looser constraints.
A technique called \textit{relaxation} is applied: A technique called \textit{relaxation} is applied:
relaxing the constraints, thereby broadening the considered domain relaxing the constraints, thereby broadening the considered domain
(e.g. by lifting the integer requirement). (e.g. by lifting the integer requirement).
@ -253,35 +251,58 @@ This consideration leads to constraints, that can be described as follows
,\end{align*}% ,\end{align*}%
\todo{Explicitly state that the first relaxation is essentially just lifing the integer \todo{Explicitly state that the first relaxation is essentially just lifing the integer
requirement}% requirement}%
where $\boldsymbol{T}_j$ is the \textit{transfer matrix}, which selects the where $\mathcal{P}_{d_j}$ is the \textit{check polytope}, the convex hull of all
binary vectors of length $d_j$ with even parity%
\footnote{Essentially $\mathcal{P}_{d_j}$ is the set of vectors that satisfy
parity-check $j$, but extended to the continuous domain.}%
and $\boldsymbol{T}_j$ is the \textit{transfer matrix}, which selects the
neighboring variable nodes neighboring variable nodes
of check node $j$ of check node $j$ (i.e., the relevant components of $\boldsymbol{c}$ for parity-check $j$).
\footnote{For example, if the $j$th row of the parity-check matrix For example, if the $j$th row of the parity-check matrix
$\boldsymbol{H}$ was $\boldsymbol{h}_j = $\boldsymbol{H}$ was $\boldsymbol{h}_j =
\begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 1 & 0 \end{bmatrix}$, \begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 1 & 0 \end{bmatrix}$,
the transfer matrix would be $\boldsymbol{T}_j = the transfer matrix would be \cite[Sec. II, A]{efficient_lp_dec_admm}
%
\begin{align*}
\boldsymbol{T}_j =
\begin{bmatrix} \begin{bmatrix}
0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
\end{bmatrix} $ (example taken from \cite[Sec. II, A]{efficient_lp_dec_admm}).} \end{bmatrix}
(i.e., the relevant components of $\boldsymbol{c}$ for parity-check $j$) .\end{align*}%
and $\mathcal{P}_{d_j}$ is the \textit{check polytope}, the convex hull of all %
binary vectors of length $d_j$ with even parity%
\footnote{Essentially $\mathcal{P}_{d_j}$ is the set of vectors that satisfy
parity-check $j$, but extended to the continuous domain.}%
.
In figure \ref{fig:dec:poly}, the two relaxations are compared for an In figure \ref{fig:dec:poly}, the two relaxations are compared for an
examplary code. examplary code, which is described by the generator and parity-check matrices%
Figure \ref{fig:dec:poly:exact} shows the codeword polytope %
$\text{poly}\left( \mathcal{C} \right) $, i.e., the constraints for the \begin{align}
equivalent linear program to exact \ac{ML} decoding - only valid codewords are \boldsymbol{G} =
feasible solutions. \begin{bmatrix}
Figure \ref{fig:dec:poly:local} shows the local codeword polytope of each check 0 & 1 & 1
node. \end{bmatrix} \label{eq:lp:example_code_def_gen} \\[1em]
Their intersection, the relaxed codeword polytope $\overline{Q}$, is shown in \boldsymbol{H} =
figure \ref{fig:dec:poly:relaxed}.% \begin{bmatrix}
1 & 1 & 1\\
0 & 1 & 1
\end{bmatrix} \label{eq:lp:example_code_def_par}
\end{align}%
%
and has only two possible codewords:
%
\begin{align*}
\mathcal{C} = \left\{ \begin{bmatrix} 0 & 0 & 0 \end{bmatrix},
\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.
The first relaxation, onto the codeword polytope $\text{poly}\left( \mathcal{C} \right) $,
is shown in figure \ref{fig:dec:poly:exact};
this constitues 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
local codeword polytopes of each check node.
% %
% %
% %
@ -295,9 +316,9 @@ figure \ref{fig:dec:poly:relaxed}.%
% Left side - codeword polytope % Left side - codeword polytope
% %
\begin{subfigure}[b]{0.49\textwidth} \begin{subfigure}[b]{0.35\textwidth}
\centering \centering
\begin{subfigure}{\textwidth} \begin{subfigure}{\textwidth}
\centering \centering
@ -305,7 +326,64 @@ figure \ref{fig:dec:poly:relaxed}.%
draw, circle, inner sep=0pt, minimum size=4pt] draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25} \tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords] \begin{tikzpicture}[scale=0.9, 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) {};
% 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) $};
\end{tikzpicture}
\caption{Set of all codewords $\mathcal{C}$}
\label{fig:dec:poly:exact_ilp}
\end{subfigure}\\[1em]
\begin{subfigure}{\textwidth}
\centering
\begin{tikzpicture}
\node (relaxation) at (0, 0) {Relaxation};
\draw (0, 0.61) -- (relaxation);
\draw[->] (relaxation) -- (0, -0.7);
\end{tikzpicture}
\vspace{4mm}
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=0.9, transform shape, tdplot_main_coords]
% Cube % Cube
\coordinate (p000) at (0, 0, 0); \coordinate (p000) at (0, 0, 0);
@ -356,12 +434,12 @@ figure \ref{fig:dec:poly:relaxed}.%
% Right side - relaxed polytope % Right side - relaxed polytope
% %
% %
\begin{subfigure}[b]{0.49\textwidth} \begin{subfigure}[b]{0.55\textwidth}
\centering \centering
\begin{subfigure}{\textwidth} \begin{subfigure}{\textwidth}
\centering \centering
\begin{minipage}{0.5\textwidth} \begin{minipage}{0.5\textwidth}
\centering \centering
@ -369,7 +447,7 @@ figure \ref{fig:dec:poly:relaxed}.%
draw, circle, inner sep=0pt, minimum size=4pt] draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25} \tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords] \begin{tikzpicture}[scale=0.9, transform shape, tdplot_main_coords]
% Cube % Cube
\coordinate (p000) at (0, 0, 0); \coordinate (p000) at (0, 0, 0);
@ -433,7 +511,7 @@ figure \ref{fig:dec:poly:relaxed}.%
draw, circle, inner sep=0pt, minimum size=4pt] draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25} \tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords] \begin{tikzpicture}[scale=0.9, transform shape, tdplot_main_coords]
% Cube % Cube
\coordinate (p000) at (0, 0, 0); \coordinate (p000) at (0, 0, 0);
@ -499,14 +577,14 @@ figure \ref{fig:dec:poly:relaxed}.%
\centering \centering
\begin{tikzpicture} \begin{tikzpicture}
\draw (-2, 0) -- (2, 0); \draw[densely dashed] (-2, 0) -- (2, 0);
\draw (-2, 0.5) -- (-2, 0); \draw[densely dashed] (-2, 0.5) -- (-2, 0);
\draw (2, 0.5) -- (2, 0); \draw[densely dashed] (2, 0.5) -- (2, 0);
\node (intersection) at (0, -0.5) {Intersection}; \node (intersection) at (0, -0.5) {Intersection};
\draw (0, 0) -- (intersection); \draw[densely dashed] (0, 0) -- (intersection);
\draw[->] (intersection) -- (0, -1); \draw[densely dashed, ->] (intersection) -- (0, -1);
\end{tikzpicture} \end{tikzpicture}
\vspace{2mm} \vspace{2mm}
@ -517,7 +595,7 @@ figure \ref{fig:dec:poly:relaxed}.%
draw, circle, inner sep=0pt, minimum size=4pt] draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25} \tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords] \begin{tikzpicture}[scale=0.9, transform shape, tdplot_main_coords]
% Cube % Cube
\coordinate (p000) at (0, 0, 0); \coordinate (p000) at (0, 0, 0);
@ -570,17 +648,25 @@ figure \ref{fig:dec:poly:relaxed}.%
\label{fig:dec:poly:relaxed} \label{fig:dec:poly:relaxed}
\end{subfigure} \end{subfigure}
\end{subfigure} \end{subfigure}
\vspace*{-2.5cm}
\hspace*{-0.1\textwidth}
\begin{tikzpicture}
\draw[->] (0,0) -- (2.5, 0);
\node[above] at (1.25, 0) {Relaxation};
% Dummy node to make tikzpicture slightly larger
\node[below] at (1.25, 0) {};
\end{tikzpicture}
\vspace{2.5cm}
\caption{Visualization of the codeword polytope and the relaxed codeword \caption{Visualization of the codeword polytope and the relaxed codeword
polytope for the code defined by the parity check matrix $\boldsymbol{H} = polytope of the code described by equations (\ref{eq:lp:example_code_def_gen})
\begin{bmatrix} and (\ref{eq:lp:example_code_def_par})}
1 & 1 & 1\\
0 & 1 & 1
\end{bmatrix}$}
\label{fig:dec:poly} \label{fig:dec:poly}
\end{figure}% \end{figure}%
% %
It can be seen that the relaxed codeword polytope $\overline{Q}$ introduces \noindent It can be seen that the relaxed codeword polytope $\overline{Q}$ introduces
vertices with fractional values; vertices with fractional values;
these represent erroneous non-codeword solutions to the linear program and these represent erroneous non-codeword solutions to the linear program and
correspond to the so-called \textit{pseudo-codewords} introduced in correspond to the so-called \textit{pseudo-codewords} introduced in