265 lines
11 KiB
TeX
265 lines
11 KiB
TeX
\chapter{Proximal Decoding}%
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\label{chapter:proximal_decoding}
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TODO
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Decoding Algorithm}%
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\label{sec:prox:Decoding Algorithm}
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Proximal decoding was proposed by Wadayama et. al as a novel formulation of
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optimization-based decoding \cite{proximal_paper}.
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With this algorithm, minimization is performed using the proximal gradient
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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 begin with the
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\ac{MAP} decoding rule, expressed as a continuous maximization problem%
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\footnote{The expansion of the domain to be continuous doesn't constitute a
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material difference in the meaning of the rule.
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The only change is that what previously were \acp{PMF} now have to be expressed
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in terms of \acp{PDF}.}
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over $\boldsymbol{x}$:%
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%
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\begin{align}
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\hat{\boldsymbol{x}} = \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}}
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f_{\tilde{\boldsymbol{X}} \mid \boldsymbol{Y}}
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\left( \tilde{\boldsymbol{x}} \mid \boldsymbol{y} \right)
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= \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}} f_{\boldsymbol{Y}
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\mid \tilde{\boldsymbol{X}}}
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\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
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f_{\tilde{\boldsymbol{X}}}\left( \tilde{\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 $f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
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\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) $ is a known function
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determined by the channel model.
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The prior \ac{PDF} $f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$ is also
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known, as the equal probability assumption is made on
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$\mathcal{C}$.
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However, since the considered domain is continuous,
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the prior \ac{PDF} cannot be ignored as a constant during the minimization
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as is often done, and has a rather unwieldy representation:%
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%
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\begin{align}
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f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right) =
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\frac{1}{\left| \mathcal{C} \right| }
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\sum_{\boldsymbol{c} \in \mathcal{C} }
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\delta\big( \tilde{\boldsymbol{x}} - \left( -1 \right) ^{\boldsymbol{c}}\big)
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\label{eq:prox:prior_pdf}
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.\end{align}%
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%
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In order to rewrite the prior \ac{PDF}
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$f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$,
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the so-called \textit{code-constraint polynomial} is introduced as:%
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%
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\begin{align*}
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h\left( \tilde{\boldsymbol{x}} \right) =
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\underbrace{\sum_{i=1}^{n} \left( \tilde{x_i}^2-1 \right) ^2}_{\text{Bipolar constraint}}
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+ \underbrace{\sum_{j=1}^{m} \left[
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\left( \prod_{i\in N_c \left( j \right) } \tilde{x_i} \right)
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-1 \right] ^2}_{\text{Parity constraint}}%
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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 one.
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In order to achieve this, the polynomial is composed of two parts: one term
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representing the bipolar 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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constraints, accommodating the role of the parity-check matrix $\boldsymbol{H}$.
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The prior \ac{PDF} is then approximated using the code-constraint polynomial as:%
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%
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\begin{align}
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f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)
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\approx \frac{1}{Z}\mathrm{e}^{-\gamma h\left( \tilde{\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 approximation in equation
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(\ref{eq:prox:prior_pdf_approx}) approaches the original function in equation
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(\ref{eq:prox:prior_pdf}).
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This approximation can then be plugged into equation (\ref{eq:prox:vanilla_MAP})
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and the likelihood can be rewritten using the negative log-likelihood
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$L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) = -\ln\left(
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f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}\left(
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\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) \right) $:%
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%
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\begin{align*}
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\hat{\boldsymbol{x}} &= \argmax_{\tilde{\boldsymbol{x}} \in \mathbb{R}^{n}}
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\mathrm{e}^{- L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) }
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\mathrm{e}^{-\gamma h\left( \tilde{\boldsymbol{x}} \right) } \\
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&= \argmin_{\tilde{\boldsymbol{x}} \in \mathbb{R}^n} \big(
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L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
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+ \gamma h\left( \tilde{\boldsymbol{x}} \right)
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\big)%
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.\end{align*}%
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%
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Thus, with proximal decoding, the objective function
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$g\left( \tilde{\boldsymbol{x}} \right)$ considered is%
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%
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\begin{align}
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g\left( \tilde{\boldsymbol{x}} \right) = L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}}
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\right)
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+ \gamma h\left( \tilde{\boldsymbol{x}} \right)%
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\label{eq:prox:objective_function}
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\end{align}%
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%
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and the decoding problem is reformulated to%
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%
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\begin{align*}
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\text{minimize}\hspace{2mm} &L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
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+ \gamma h\left( \tilde{\boldsymbol{x}} \right)\\
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\text{subject to}\hspace{2mm} &\tilde{\boldsymbol{x}} \in \mathbb{R}^n
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.\end{align*}
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%
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For the solution of the approximate \ac{MAP} decoding problem, the two parts
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of equation (\ref{eq:prox:objective_function}) are considered separately:
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the minimization of the objective function occurs in an alternating
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fashion, switching between the negative log-likelihood
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$L\left( \boldsymbol{y} \mid \boldsymbol{x} \right) $ and the scaled
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code-constraint 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, is performed using gradient
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descent:%
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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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\label{eq:prox:step_log_likelihood}
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.\end{align}%
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%
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For the second step, minimizing the scaled code-constraint polynomial, the
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proximal gradient method is used and the \textit{proximal operator} of
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$\gamma h\left( \tilde{\boldsymbol{x}} \right) $ has to be computed.
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It is then immediately approximated with gradient-descent:%
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%
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\begin{align*}
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\textbf{prox}_{\gamma h} \left( \tilde{\boldsymbol{x}} \right) &\equiv
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\argmin_{\boldsymbol{t} \in \mathbb{R}^n}
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\left( \gamma h\left( \boldsymbol{t} \right) +
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\frac{1}{2} \lVert \boldsymbol{t} - \tilde{\boldsymbol{x}} \rVert \right)\\
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&\approx \tilde{\boldsymbol{x}} - \gamma \nabla h \left( \tilde{\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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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 \nabla h\left( \boldsymbol{r} \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 approximation of the prior \ac{PDF} made in equation (\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( \tilde{\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.'' \cite[Sec. 3.1]{proximal_paper}.
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%The components of the gradient of the code-constraint polynomial can be computed as follows:%
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%%
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%\begin{align*}
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% \frac{\partial}{\partial x_k} h\left( \boldsymbol{x} \right) =
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% 4\left( x_k^2 - 1 \right) x_k + \frac{2}{x_k}
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% \sum_{i\in \mathcal{B}\left( k \right) } \left(
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% \left( \prod_{j\in\mathcal{A}\left( i \right)} x_j\right)^2
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% - \prod_{j\in\mathcal{A}\left( i \right) }x_j \right)
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%.\end{align*}%
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%\todo{Only multiplication?}%
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%\todo{$x_k$: $k$ or some other indexing variable?}%
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%%
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In the case of \ac{AWGN}, the likelihood
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$f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
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\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)$
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is%
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%
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\begin{align*}
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f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
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\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
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= \frac{1}{\sqrt{2\pi\sigma^2}}\mathrm{e}^{
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-\frac{\lVert \boldsymbol{y}-\tilde{\boldsymbol{x}}
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\rVert^2 }
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{2\sigma^2}}
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.\end{align*}
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%
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Thus, the gradient of the negative log-likelihood becomes%
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\footnote{For the minimization, constants can be disregarded. For this reason,
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it suffices to consider only proportionality instead of equality.}%
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%
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\begin{align*}
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\nabla L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
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&\propto -\nabla \lVert \boldsymbol{y} - \tilde{\boldsymbol{x}} \rVert^2\\
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&\propto \tilde{\boldsymbol{x}} - \boldsymbol{y}
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,\end{align*}%
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%
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allowing equation (\ref{eq:prox:step_log_likelihood}) to be rewritten as%
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%
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\begin{align*}
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\boldsymbol{r} \leftarrow \boldsymbol{s}
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- \omega \left( \boldsymbol{s} - \boldsymbol{y} \right)
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.\end{align*}
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%
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One thing to consider during the actual decoding process, is that the gradient
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of the code-constraint polynomial can take on extremely large values.
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To avoid numerical instability, an additional step is added, where all
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components of the current estimate are clipped to $\left[-\eta, \eta \right]$,
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where $\eta$ is a positive constant slightly larger than one:%
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%
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\begin{align*}
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\boldsymbol{s} \leftarrow \Pi_{\eta} \left( \boldsymbol{r}
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- \gamma \nabla h\left( \boldsymbol{r} \right) \right)
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,\end{align*}
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%
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$\Pi_{\eta}\left( \cdot \right) $ expressing the projection onto
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$\left[ -\eta, \eta \right]^n$.
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The iterative decoding process resulting from these considerations 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 \left( \boldsymbol{s} - \boldsymbol{y} \right) $
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$\boldsymbol{s} \leftarrow \Pi_\eta \left(\boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) \right)$
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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 for an \ac{AWGN} channel}
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\label{fig:prox:alg}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Implementation Details}%
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\label{sec:prox:Implementation Details}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results}%
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\label{sec:prox:Results}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Improved Implementation}%
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\label{sec:prox:Improved Implementation}
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