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Split LP and ADMM into two sections; done with cost function derivation for LP

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Andreas Tsouchlos 2023-02-18 16:58:35 +01:00
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latex/thesis/chapters/decoding_techniques.tex
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@ -160,8 +160,8 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{LP Decoding using ADMM}% \section{LP Decoding}%
\label{sec:dec:LP Decoding using ADMM} \label{sec:dec:LP Decoding}
\Ac{LP} decoding is a subject area introduced by Feldman et al. \Ac{LP} decoding is a subject area introduced by Feldman et al.
\todo{Space before citation?} \todo{Space before citation?}
@ -171,40 +171,40 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
decoding and one, which is an approximation with a more manageable decoding and one, which is an approximation with a more manageable
representation. representation.
To solve the resulting linear program, various optimization methods can be To solve the resulting linear program, various optimization methods can be
used; used.
the one examined in this work is \ac{ADMM}.
\todo{Why chose ADMM?}
Feldman at al. begin by looking at the \ac{ML} decoding problem% Feldman at al. begin by looking at the \ac{ML} decoding problem%
\footnote{They assume that all codewords are equally likely to be transmitted, \footnote{They assume that all codewords are equally likely to be transmitted,
making the \ac{ML} and \ac{MAP} decoding problems equivalent.}% making the \ac{ML} and \ac{MAP} decoding problems equivalent.}%
% %
\begin{align*} \begin{align}
\hat{\boldsymbol{c}} = \argmax_{\boldsymbol{c} \in \mathcal{C}} \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}%
%
Assuming a memoryless channel, \ref{eq:lp:ml} can be rewritten in terms
of the \acp{LLR} $\gamma_i$ \cite[Sec 2.5]{feldman_thesis}:%
%
\begin{align*}
\hat{\boldsymbol{c}} = \argmin_{\boldsymbol{c}\in\mathcal{C}}
\sum_{i=1}^{n} \gamma_i y_i,%
\hspace{5mm} \gamma_i = \ln\left(
\frac{f_{\boldsymbol{Y} | \boldsymbol{C}}
\left( Y_i = y_i \mid C_i = 0 \right) }
{f_{\boldsymbol{Y} | \boldsymbol{C}}
\left( Y_i = y_i | C_i = 1 \right) } \right)
.\end{align*} .\end{align*}
% %
They suggest that maximizing the likelihood The authors propose the following cost function%
$f_{\boldsymbol{Y} \mid \boldsymbol{C}}\left( \boldsymbol{y} \mid \boldsymbol{c} \right)$
is equivalent to minimizing the negative log-likelihood.
\ldots (Explain arriving at the cost function from the ML decoding problem)
Based on this, they propose their cost function%
\footnote{In this context, \textit{cost function} and \textit{objective function} \footnote{In this context, \textit{cost function} and \textit{objective function}
have the same meaning.} have the same meaning.}
for the \ac{LP} decoding problem:% for the \ac{LP} decoding problem:%
% %
\begin{align*} \begin{align*}
\sum_{i=1}^{n} \gamma_i c_i, \sum_{i=1}^{n} \gamma_i c_i
\hspace{5mm} \gamma_i = \ln\left(
\frac{f_{\boldsymbol{Y} | \boldsymbol{C}}
\left( Y_i = y_i \mid C_i = 0 \right) }
{f_{\boldsymbol{Y} | \boldsymbol{C}}
\left( Y_i = y_i | C_i = 1 \right) } \right)
.\end{align*} .\end{align*}
% %
%
With this cost function, the exact integer linear program formulation of \ac{ML} With this cost function, the exact integer linear program formulation of \ac{ML}
decoding is the following:% decoding is the following:%
% %
@ -213,6 +213,8 @@ decoding is the following:%
\text{subject to }\hspace{2mm} &\boldsymbol{c} \in \mathcal{C} \text{subject to }\hspace{2mm} &\boldsymbol{c} \in \mathcal{C}
.\end{align*}% .\end{align*}%
% %
\todo{$\boldsymbol{c}$ or some other variable name? e.g. $\boldsymbol{c}^{*}$.
Especially for the continuous consideration 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 one with looser constraints.
@ -551,11 +553,27 @@ vertices;
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{pseudocodewords} introduced in correspond to the so-called \textit{pseudocodewords} introduced in
\cite{feldman_paper}. \cite{feldman_paper}.
However, since for \ac{LDPC} codes $Q$ scales linearly with $n$, it is a lot However, since for \ac{LDPC} codes $Q$ scales linearly with $n$ instead of
more tractable for practical applications. exponentially, it is a lot more tractable for practical applications.
The resulting formulation of the relaxed optimization problem
(called \ac{LCLP} by the authors) is the following:%
%
\begin{align*}
\text{minimize }\hspace{2mm} &\sum_{i=1}^{n} \gamma_i c_i \\
\text{subject to }\hspace{2mm} &\ldots
.\end{align*}%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{LP Decoding using ADMM}%
\label{sec:dec:LP Decoding using ADMM}
\begin{itemize} \begin{itemize}
\item TODO: \Ac{ADMM} as a solver \item Why ADMM?
\item Adaptive Linear Programming?
\item How ADMM is adapted to LP decoding
\end{itemize} \end{itemize}