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Reworked introduction to decoding using optimization

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Andreas Tsouchlos 2023-02-19 11:02:33 +01:00
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latex/thesis/chapters/decoding_techniques.tex
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@ -30,25 +30,24 @@ the \ac{ML} decoding problem:%
f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c} \right)
.\end{align*}%
%
\todo{Note about these generally being the same thing, when the a priori probability
is uniformly distributed}%
\todo{Here the two problems are written in terms of $\hat{\boldsymbol{c}}$; below MAP
decoding is applied in terms of $\hat{\boldsymbol{x}}$. Is that a problem?}%
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*}
\text{minimize } f\left( \boldsymbol{c} \right)\\
\text{subject to $\boldsymbol{c} \in D$}
,\end{align*}%
%
where $D$ is the domain of values attainable for $c$ and represents the
constraints.
In contrast to the established message-passing decoding algorithms,
the viewpoint then changes from observing the decoding process in its
tanner graph representation (as shown in figure \ref{fig:dec:tanner})
to 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}).
to a spacial representation (figure \ref{fig:dec:spacial}),
where the codewords are some of the edges of a hypercube.
The goal is to find that point $\boldsymbol{c}$,
which minimizes the objective function $f$.
%
% Figure showing decoding space
@ -143,11 +142,11 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
\node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
% x
% c
\node[color=KITgreen, fill=KITgreen,
draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.9, 0.7, 1) {};
\node[color=KITgreen, right=0cm of f] {$\boldsymbol{x}$};
draw, circle, inner sep=0pt, minimum size=4pt] (c) at (0.9, 0.7, 1) {};
\node[color=KITgreen, right=0cm of c] {$\boldsymbol{c}$};
\end{tikzpicture}
\caption{Spacial representation of a single parity-check code}
@ -164,7 +163,6 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
\label{sec:dec:LP Decoding}
\Ac{LP} decoding is a subject area introduced by Feldman et al.
\todo{Space before citation?}
\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}
@ -179,7 +177,8 @@ making the \ac{ML} and \ac{MAP} decoding problems equivalent.}%
%
\begin{align}
\hat{\boldsymbol{c}} = \argmax_{\boldsymbol{c} \in \mathcal{C}}
f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c} \right)%
f_{\boldsymbol{Y} \mid \boldsymbol{C}}
\left( \boldsymbol{y} \mid \boldsymbol{c} \right)%
\label{eq:lp:ml}
.\end{align}%
%