From 40e731d1117053f99c8d6e8ad2d769aeaea24797 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Sat, 18 Feb 2023 16:58:35 +0100
Subject: [PATCH] Split LP and ADMM into two sections; done with cost function
 derivation for LP

---
 latex/thesis/chapters/decoding_techniques.tex | 66 ++++++++++++-------
 1 file changed, 42 insertions(+), 24 deletions(-)

diff --git a/latex/thesis/chapters/decoding_techniques.tex b/latex/thesis/chapters/decoding_techniques.tex
index 7303ab6..68a5e7b 100644
--- a/latex/thesis/chapters/decoding_techniques.tex
+++ b/latex/thesis/chapters/decoding_techniques.tex
@@ -160,8 +160,8 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{LP Decoding using ADMM}%
-\label{sec:dec:LP Decoding using ADMM}
+\section{LP Decoding}%
+\label{sec:dec:LP Decoding}
 
 \Ac{LP} decoding is a subject area introduced by Feldman et al.
 \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
 representation.
 To solve the resulting linear program, various optimization methods can be
-used;
-the one examined in this work is \ac{ADMM}.
-\todo{Why chose ADMM?}
+used.
 
 Feldman at al. begin by looking at the \ac{ML} decoding problem%
 \footnote{They assume that all codewords are equally likely to be transmitted,
 making the \ac{ML} and \ac{MAP} decoding problems equivalent.}%
 %
-\begin{align*}
+\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}%
+%
+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*}
 %
-They suggest that maximizing the likelihood
-$f_{\boldsymbol{Y} \mid \boldsymbol{C}}\left( \boldsymbol{y} \mid \boldsymbol{c} \right)$
-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%
+The authors propose the following cost function%
 \footnote{In this context, \textit{cost function} and \textit{objective function}
 have the same meaning.}
 for the \ac{LP} decoding problem:%
 %
 \begin{align*}
-    \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)
+    \sum_{i=1}^{n} \gamma_i c_i
 .\end{align*}
 %
-%
 With this cost function, the exact integer linear program formulation of \ac{ML}
 decoding is the following:%
 %
@@ -213,6 +213,8 @@ decoding is the following:%
     \text{subject to }\hspace{2mm} &\boldsymbol{c} \in \mathcal{C}
 .\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
 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
 correspond to the so-called \textit{pseudocodewords} introduced in
 \cite{feldman_paper}.
-However, since for \ac{LDPC} codes $Q$ scales linearly with $n$, it is a lot
-more tractable for practical applications.
+However, since for \ac{LDPC} codes $Q$ scales linearly with $n$ instead of
+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}
-    \item TODO: \Ac{ADMM} as a solver
+    \item Why ADMM?
+    \item Adaptive Linear Programming?
+    \item How ADMM is adapted to LP decoding
 \end{itemize}