233 lines
8.9 KiB
TeX
233 lines
8.9 KiB
TeX
\chapter{Decoding Techniques}%
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\label{chapter:decoding_techniques}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Decoding using Optimization Methods}%
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\label{sec:dec:Decoding using Optimization Methods}
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%
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% TODOs
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%
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\begin{itemize}
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\item General methodology
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\end{itemize}
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%
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% Figure showing decoding space
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%
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\begin{figure}[H]
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{245}
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\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
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% Cube
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\draw[dashed] (0, 0, 0) -- (2, 0, 0);
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\draw[dashed] (2, 0, 0) -- (2, 0, 2);
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\draw[] (2, 0, 2) -- (0, 0, 2);
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\draw[] (0, 0, 2) -- (0, 0, 0);
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\draw[] (0, 2, 0) -- (2, 2, 0);
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\draw[] (2, 2, 0) -- (2, 2, 2);
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\draw[] (2, 2, 2) -- (0, 2, 2);
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\draw[] (0, 2, 2) -- (0, 2, 0);
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\draw[] (0, 0, 0) -- (0, 2, 0);
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\draw[dashed] (2, 0, 0) -- (2, 2, 0);
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\draw[] (2, 0, 2) -- (2, 2, 2);
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\draw[] (0, 0, 2) -- (0, 2, 2);
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% Polytope Annotations
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\node[codeword] (c000) at (0, 0, 0) {};% {$\left( 0, 0, 0 \right) $};
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\node[codeword] (c101) at (2, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
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\node[codeword] (c110) at (2, 2, 0) {};% {$\left( 1, 1, 0 \right) $};
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\node[codeword] (c011) at (0, 2, 2) {};% {$\left( 0, 1, 1 \right) $};
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\node[color=KITblue, right=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c101] {$\left( 1, 0, 1 \right) $};
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\node[color=KITblue, left=0cm of c110] {$\left( 1, 1, 0 \right) $};
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\node[color=KITblue, left=-0.1cm of c011] {$\left( 0, 1, 1 \right) $};
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% f
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\node[color=KITgreen, fill=KITgreen,
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draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.9, 0.7, 1) {};
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\node[color=KITgreen, right=0cm of f] {$\boldsymbol{f}$};
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\end{tikzpicture}
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\caption{Hypercube ($n=3$) and valid codewords for a single parity-check code}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{LP Decoding using ADMM}%
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\label{sec:dec:LP Decoding using ADMM}
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\begin{itemize}
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\item Equivalent \ac{ML} optimization problem
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\item \Ac{LP} relaxation
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\item \Ac{ADMM} as a solver
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Proximal Decoding}%
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\label{sec:dec:Proximal Decoding}
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Proximal decoding was proposed by Wadayama et. al \cite{proximal_paper}.
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With this decoding algorithm, the objective function is minimized using
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the proximal gradient method.
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In contrast to \ac{LP} decoding, the objective function is based on a
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non-convex optimization formulation of the \ac{MAP} decoding problem.
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In order to derive the objective function, the authors reformulate the
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\ac{MAP} decoding problem:%
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%
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\begin{align}
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\hat{\boldsymbol{x}} = \argmax_{\boldsymbol{x} \in \mathbb{R}^{n}}
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f_{\boldsymbol{X} \mid \boldsymbol{Y}}
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\left( \boldsymbol{x} \mid \boldsymbol{y} \right)
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= \argmax_{\boldsymbol{x} \in \mathbb{R}^{n}} f_{\boldsymbol{Y} \mid \boldsymbol{X}}
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\left( \boldsymbol{y} \mid \boldsymbol{x} \right)
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f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)%
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\label{eq:prox:vanilla_MAP}
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\end{align}%
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%
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The likelihood is usually a known function determined by the channel model.
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In order to rewrite the prior \ac{PDF}
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$f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)$,
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the so-called \textit{code-constraint polynomial} is introduced:%
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%
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\begin{align}
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h\left( \boldsymbol{x} \right) = \sum_{j=1}^{n} \left( x_j^2-1 \right) ^2
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+ \sum_{i=1}^{m} \left[
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\left( \prod_{j\in \mathcal{A}\left( i \right) } x_j \right) -1 \right] ^2%
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\label{eq:prox:ccp}
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\end{align}%
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%
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The intention of this function is to provide a way to penalize vectors far
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from a codeword and favor those close to a codeword.
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In order to achieve this, the polynomial is composed of two parts: one term
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representing the bibolar constraint, providing for a discrete solution of the
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continuous optimization problem, and one term representing the parity
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constraint, accomodating the role of the parity-check matrix $\boldsymbol{H}$.
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%
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The prior \ac{PDF} is then approximated using the code-constraint polynomial\todo{Italic?}:%
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%
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\begin{align}
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f_{\boldsymbol{X}}\left( \boldsymbol{x} \right) =
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\frac{1}{\left| \mathcal{C}\left( \boldsymbol{H} \right) \right| }
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\sum_{c \in \mathcal{C}\left( \boldsymbol{H} \right) }
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\delta\left( \boldsymbol{x} - \left( -1 \right) ^{\boldsymbol{c}}\right)
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\approx \frac{1}{Z}e^{-\gamma h\left( \boldsymbol{x} \right) }%
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\label{eq:prox:prior_pdf_approx}
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\end{align}%
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%
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The authors justify this approximation by arguing that for
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$\gamma \rightarrow \infty$, the right-hand side aproaches the left-hand
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side. In \ref{eq:prox:vanilla_MAP} the prior \ac{PDF}
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$f_{\boldsymbol{X}}\left( \boldsymbol{x} \right) $ can then be subsituted
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for \ref{eq:prox:prior_pdf_approx} and the likelihood can be rewritten using
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the negative log-likelihood
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$f_{\boldsymbol{X} \mid \boldsymbol{Y}}\left( \boldsymbol{x} \mid \boldsymbol{y} \right)
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= e^{- L\left( \boldsymbol{y} \mid \boldsymbol{x} \right) }$:%
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%
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\begin{align}
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\hat{\boldsymbol{x}} &= \argmax_{\boldsymbol{x} \in \mathbb{R}^{n}}
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e^{- L\left( \boldsymbol{y} \mid \boldsymbol{x} \right) }
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e^{-\gamma h\left( \boldsymbol{x} \right) } \nonumber \\
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&= \argmin_{\boldsymbol{x} \in \mathbb{R}^n} \left(
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L\left( \boldsymbol{y} \mid \boldsymbol{x} \right)
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+ \gamma h\left( \boldsymbol{x} \right)
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\right)%
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\label{eq:prox:approx_map_problem}
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\end{align}%
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%
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Thus, with proximal decoding, the objective function
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$f\left( \boldsymbol{x} \right)$ to be minimized is%
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%
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\begin{align}
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f\left( \boldsymbol{x} \right) = L\left( \boldsymbol{x} \mid \boldsymbol{y} \right)
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+ \gamma h\left( \boldsymbol{x} \right)%
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\label{eq:prox:objective_function}
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.\end{align}\todo{Dot after equations?}
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For the solution of the approximalte \ac{MAP} decoding problem, the two parts
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of \ref{eq:prox:approx_map_problem} are considered separately from one
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another: the minimization of the objective function occurs in an alternating
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manner, switching between the minimization of the negative log-likelihood
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$L\left( \boldsymbol{y} \mid \boldsymbol{x} \right) $ and the scaled
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code-constaint polynomial $\gamma h\left( \boldsymbol{x} \right) $.
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Two helper variables, $\boldsymbol{r}$ and $\boldsymbol{s}$ are introduced,
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describing the result of each of the two steps.
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The first step, minimizing the log-likelihood using gradient descent, yields%
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%
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\begin{align*}
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\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \nabla
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L\left( \boldsymbol{y} \mid \boldsymbol{s} \right),
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\hspace{5mm}\omega > 0
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.\end{align*}%
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%
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For the second step, minimizig the scaled code-constraint polynomial using
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the proximal gradient method, the proximal operator of
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$\gamma h\left( \boldsymbol{x} \right) $ has to be computed and is
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immediately approximalted by a gradient-descent step:%
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%
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\begin{align*}
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\text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv
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\argmin_{\boldsymbol{t} \in \mathbb{R}^n}
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\left( \gamma h\left( \boldsymbol{x} \right) +
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\frac{1}{2} \lVert \boldsymbol{t} - \boldsymbol{x} \rVert \right)\\
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&\approx \boldsymbol{x} - \gamma h \left( \boldsymbol{x} \right),
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\hspace{5mm} \gamma \text{ small}
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.\end{align*}%
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%
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The second step thus becomes%
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%
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\begin{align*}
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\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma h\left( \boldsymbol{x} \right),
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\hspace{5mm}\gamma > 0,\text{ small}
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.\end{align*}
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%
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While the approximatin of the prior \ac{PDF} made in \ref{eq:prox:prior_pdf_approx}
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theoretically becomes better
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with larger $\gamma$, the constraint that $\gamma$ be small is important,
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as it keeps the effect of $h\left( \boldsymbol{x} \right) $ on the landscape
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of the objective function small.
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Otherwise, unwanted stationary points, including local minima are introduced.
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The authors say that in practice, the value of $\gamma$ should be adjusted
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according to the decoding performance.
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The iterative decoding process resulting from this considreation is shown in
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figure \ref{fig:prox:alg}.
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\begin{figure}[H]
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\centering
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\begin{genericAlgorithm}[caption={}, label={}]
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$\boldsymbol{s} \leftarrow \boldsymbol{0}$
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for $K$ iterations do
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$\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \nabla L \left( \boldsymbol{y} \mid \boldsymbol{s} \right) $
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$\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) $
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$\boldsymbol{\hat{x}} \leftarrow \text{sign}\left( \boldsymbol{s} \right) $
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if $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ do
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return $\boldsymbol{\hat{c}}$
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end if
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end for
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return $\boldsymbol{\hat{c}}$
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\end{genericAlgorithm}
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\caption{Proximal decoding algorithm}
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\label{fig:prox:alg}
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\end{figure}
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