From 364a85248f7912d0dd1c83c2b5d12a2c416e817d Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Mon, 23 Jan 2023 20:42:54 +0100
Subject: [PATCH] Added comparison between proximal and hybrid decoding

---
 .../midterm/sections/examination_results.tex  | 61 +++++++++++++++++++
 1 file changed, 61 insertions(+)

diff --git a/latex/presentations/midterm/sections/examination_results.tex b/latex/presentations/midterm/sections/examination_results.tex
index a545282..bd82517 100644
--- a/latex/presentations/midterm/sections/examination_results.tex
+++ b/latex/presentations/midterm/sections/examination_results.tex
@@ -347,6 +347,67 @@ Output $\boldsymbol{\tilde{x}}_n$ with lowest $d\left( \boldsymbol{ \tilde{x}}_n
 \end{frame}
 
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[t, fragile]
+    \frametitle{Proximal Decoding: Improvement using ``ML-on-List''}
+    \setcounter{footnote}{0} 
+
+    \begin{itemize}
+        \item Improvement of proximal decoding by adding an ``ML-on-list'' step after iterating
+    \end{itemize}
+
+    \begin{minipage}[t]{.48\textwidth}
+        \centering
+
+        \begin{figure}
+           \centering
+
+            \begin{algorithm}[caption={}, label={},
+            basicstyle=\fontsize{7.5}{9.5}\selectfont
+            ]
+$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$
+for $k=0$ to $K-1$ do
+    $\boldsymbol{r}^{\left( k+1 \right)} = \boldsymbol{s}^{(k)} - \omega \nabla L \left( \boldsymbol{s}^{(k)}; \boldsymbol{y} \right) $
+    Compute $\nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)$
+    $\boldsymbol{s}^{\left( k+1 \right)} = \boldsymbol{r}^{(k+1)} - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right) $
+    $\boldsymbol{\hat{x}} = \text{sign}\left( \boldsymbol{s}^{\left( k+1 \right) } \right) $
+    If $\boldsymbol{\hat{x}}$ passes the parity check condition, break the loop.
+end for
+Output $\boldsymbol{\hat{x}}$
+            \end{algorithm}
+
+            \caption{Proximal decoding algorithm}
+        \end{figure}
+
+    \end{minipage}%
+    \hfill\begin{minipage}[t]{.48\textwidth}
+        \centering
+        \begin{figure}
+           \centering 
+
+            \begin{algorithm}[caption={}, label={},
+            basicstyle=\fontsize{7.5}{9.5}\selectfont
+            ]
+$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$
+for $k=0$ to $K-1$ do
+    $\boldsymbol{r}^{\left( k+1 \right)} = \boldsymbol{s}^{(k)} - \omega \nabla L \left( \boldsymbol{s}^{(k)}; \boldsymbol{y} \right) $
+    Compute $\nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)$
+    $\boldsymbol{s}^{\left( k+1 \right)} = \boldsymbol{r}^{(k+1)} - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right) $
+    $\boldsymbol{\hat{x}} = \text{sign}\left( \boldsymbol{s}^{\left( k+1 \right) } \right) $
+    $\textcolor{KITblue}{\text{If }\boldsymbol{\hat{x}}\text{ passes the parity check condition, output }\boldsymbol{\hat{x}}}$
+end for
+$\textcolor{KITblue}{\text{Find }N\text{ most probably wrong bits.}}$
+$\textcolor{KITblue}{\text{Generate variations } \boldsymbol{\tilde{x}}_n\text{ of }\boldsymbol{\hat{x}}\text{ with the }N\text{ bits modified.}}$
+$\textcolor{KITblue}{\text{Compute }d_H\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right) \forall n \in \left[ 1 : N-1 \right]}$
+$\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{x}}_n\text{ with lowest }d_H\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)}$
+            \end{algorithm}
+
+            \caption{Hybrid proximal \& ML decoding algorithm}
+        \end{figure}
+    \end{minipage} 
+\end{frame}
+
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{ADMM: Examination Results}%
 \label{sub:Ex ADMM}