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Fixed typos; Added citation

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Andreas Tsouchlos 2023-03-08 13:55:20 +01:00
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latex/thesis/chapters/decoding_techniques.tex
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@ -189,7 +189,7 @@ making the \ac{ML} and \ac{MAP} decoding problems equivalent.}%
.\end{align}% .\end{align}%
% %
Assuming a memoryless channel, equation (\ref{eq:lp:ml}) can be rewritten in terms Assuming a memoryless channel, equation (\ref{eq:lp:ml}) can be rewritten in terms
of the \acp{LLR} $\gamma_i$ \cite[Sec 2.5]{feldman_thesis}:% of the \acp{LLR} $\gamma_i$ \cite[Sec. 2.5]{feldman_thesis}:%
% %
\begin{align*} \begin{align*}
\hat{\boldsymbol{c}} = \argmin_{\boldsymbol{c}\in\mathcal{C}} \hat{\boldsymbol{c}} = \argmin_{\boldsymbol{c}\in\mathcal{C}}
@ -236,6 +236,7 @@ decoding, redefining the constraints in terms of the \text{codeword polytope}
% %
which represents the \textit{convex hull} of all possible codewords, which represents the \textit{convex hull} of all possible codewords,
i.e., the convex set of linear combinations of all codewords. i.e., the convex set of linear combinations of all codewords.
This corresponds to simply lifting the integer requirement.
However, since the number of constraints needed to characterize the codeword However, since the number of constraints needed to characterize the codeword
polytope is exponential in the code length, this formulation is relaxed further. polytope is exponential in the code length, this formulation is relaxed further.
By observing that each check node defines its own local single parity-check By observing that each check node defines its own local single parity-check
@ -249,8 +250,7 @@ This consideration leads to constraints, that can be described as follows
\boldsymbol{T}_j \tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j} \boldsymbol{T}_j \tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}
\hspace{5mm}\forall j\in \mathcal{J} \hspace{5mm}\forall j\in \mathcal{J}
,\end{align*}% ,\end{align*}%
\todo{Explicitly state that the first relaxation is essentially just lifing the integer %
requirement}%
where $\mathcal{P}_{d_j}$ is the \textit{check polytope}, the convex hull of all where $\mathcal{P}_{d_j}$ is the \textit{check polytope}, the convex hull of all
binary vectors of length $d_j$ with even parity% binary vectors of length $d_j$ with even parity%
\footnote{Essentially $\mathcal{P}_{d_j}$ is the set of vectors that satisfy \footnote{Essentially $\mathcal{P}_{d_j}$ is the set of vectors that satisfy
@ -827,7 +827,7 @@ It is then immediately approximated with gradient-descent:%
\begin{align*} \begin{align*}
\text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv \text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv
\argmin_{\boldsymbol{t} \in \mathbb{R}^n} \argmin_{\boldsymbol{t} \in \mathbb{R}^n}
\left( \gamma h\left( \boldsymbol{x} \right) + \left( \gamma h\left( \boldsymbol{t} \right) +
\frac{1}{2} \lVert \boldsymbol{t} - \boldsymbol{x} \rVert \right)\\ \frac{1}{2} \lVert \boldsymbol{t} - \boldsymbol{x} \rVert \right)\\
&\approx \boldsymbol{r} - \gamma \nabla h \left( \boldsymbol{r} \right), &\approx \boldsymbol{r} - \gamma \nabla h \left( \boldsymbol{r} \right),
\hspace{5mm} \gamma > 0, \text{ small} \hspace{5mm} \gamma > 0, \text{ small}
@ -847,7 +847,7 @@ as it keeps the effect of $h\left( \boldsymbol{x} \right) $ on the landscape
of the objective function small. of the objective function small.
Otherwise, unwanted stationary points, including local minima, are introduced. Otherwise, unwanted stationary points, including local minima, are introduced.
The authors say that in practice, the value of $\gamma$ should be adjusted The authors say that in practice, the value of $\gamma$ should be adjusted
according to the decoding performance. according to the decoding performance \cite[Sec. 3.1]{proximal_paper}.
%The components of the gradient of the code-constraint polynomial can be computed as follows:% %The components of the gradient of the code-constraint polynomial can be computed as follows:%
%% %%