From 1bc78b0675aa4e1e9652a386dc51762d0cfcc44a Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Mon, 19 Dec 2022 03:36:57 +0100
Subject: [PATCH] Added LP Decoding slide

---
 latex/presentations/midterm/presentation.tex  |  1 +
 .../sections/theoretical_background.tex       | 83 ++++++++++++++++++-
 2 files changed, 83 insertions(+), 1 deletion(-)

diff --git a/latex/presentations/midterm/presentation.tex b/latex/presentations/midterm/presentation.tex
index 0b85568..80b304b 100644
--- a/latex/presentations/midterm/presentation.tex
+++ b/latex/presentations/midterm/presentation.tex
@@ -6,6 +6,7 @@
 %\usepackage{amsmath}
 \usepackage{float}
 \usepackage{tikz}
+\usepackage{tikz-3dplot}
 \usepackage{pgfplots}
 \pgfplotsset{compat=newest}
 \usepackage{subcaption}
diff --git a/latex/presentations/midterm/sections/theoretical_background.tex b/latex/presentations/midterm/sections/theoretical_background.tex
index ea5727a..4c4c976 100644
--- a/latex/presentations/midterm/sections/theoretical_background.tex
+++ b/latex/presentations/midterm/sections/theoretical_background.tex
@@ -18,7 +18,7 @@
 \label{sub:Presumptions}
 
 \begin{frame}[t]
-    \frametitle{Channel \& Modulation}
+    \frametitle{Presumptions: Channel \& Modulation}
 
     \tikzstyle{mapper} = [rectangle, minimum width=1.5cm, rounded corners=0.1cm, minimum height=0.7cm, text centered, draw=black, fill=KITgreen!80]
     \begin{figure}[htpb]
@@ -64,6 +64,87 @@
 \begin{frame}[t]
     \frametitle{LP Decoding}
 
+    \begin{minipage}[c]{0.6\linewidth}
+        \begin{itemize}
+            \item Codeword Polytope:
+                \begin{align*}
+                    \text{poly}\left( \mathcal{C} \right) = \left\{ \sum_{\boldsymbol{y}\in\mathcal{C}} \lambda_{\boldsymbol{y}} \boldsymbol{y} : \lambda_{\boldsymbol{y}} \ge  0, \sum_{\boldsymbol{y}\in\mathcal{C}}\lambda_{\boldsymbol{y}} = 1 \right\}, \hspace{5mm} \lambda_{\boldsymbol{y}} \in \mathbb{R}
+                \end{align*}
+            \item Cost Function:
+                \begin{align*}
+                    \gamma_i = \log\left( \frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
+                \end{align*}
+            \item LP Formulation:
+                \begin{align*}
+                    &\text{minimize } \sum_{i=1}^{n} \gamma_i f_i, \hspace{5mm} f_i = \sum_{\boldsymbol{y}} \lambda_{\boldsymbol{y}}y_i\\
+                    &\text{subject to } \boldsymbol{f}\in\text{poly}\left( \mathcal{C} \right) 
+                \end{align*}
+        \end{itemize}
+    \end{minipage}%
+    \hfill%
+    \begin{minipage}[c]{0.4\linewidth}
+        \begin{figure}[H]
+            \centering
+
+            \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
+                                    draw, circle, inner sep=0pt, minimum size=4pt]
+
+            \tdplotsetmaincoords{60}{245}
+            \begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
+                % Cube
+
+                \draw[dashed] (0, 0, 0) -- (2, 0, 0);
+                \draw[dashed] (2, 0, 0) -- (2, 0, 2);
+                \draw[] (2, 0, 2) -- (0, 0, 2);
+                \draw[] (0, 0, 2) -- (0, 0, 0);
+
+                \draw[] (0, 2, 0) -- (2, 2, 0);
+                \draw[] (2, 2, 0) -- (2, 2, 2);
+                \draw[] (2, 2, 2) -- (0, 2, 2);
+                \draw[] (0, 2, 2) -- (0, 2, 0);
+
+                \draw[] (0, 0, 0) -- (0, 2, 0);
+                \draw[dashed] (2, 0, 0) -- (2, 2, 0);
+                \draw[] (2, 0, 2) -- (2, 2, 2);
+                \draw[] (0, 0, 2) -- (0, 2, 2);
+
+                % Codeword Polytope
+
+                \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 0, 2);
+                \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 2, 0);
+                \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (0, 2, 2);
+
+                \draw[line width=1pt, color=KITblue] (2, 0, 2) -- (2, 2, 0);
+                \draw[line width=1pt, color=KITblue] (2, 0, 2) -- (0, 2, 2);
+
+                \draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);
+
+                % Polytope Annotations
+
+                \node[codeword] (c000) at (0, 0, 0) {};% {$\left( 0, 0, 0 \right) $};
+                \node[codeword] (c101) at (2, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
+                \node[codeword] (c110) at (2, 2, 0) {};% {$\left( 1, 1, 0 \right) $};
+                \node[codeword] (c011) at (0, 2, 2) {};% {$\left( 0, 1, 1 \right) $};
+
+                \node[color=KITblue, right=0cm of c000] {$\left( 0, 0, 0 \right) $};
+                \node[color=KITblue, above=0cm of c101] {$\left( 1, 0, 1 \right) $};
+                \node[color=KITblue, left=0cm of c110] {$\left( 1, 1, 0 \right) $};
+                \node[color=KITblue, left=0cm of c011] {$\left( 0, 1, 1 \right) $};
+
+                % f
+
+                \node[color=KITgreen, fill=KITgreen,
+                      draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.7, 0.7, 1) {};
+                \node[color=KITgreen, right=0cm of f] {$\boldsymbol{f}$};
+            \end{tikzpicture}
+            \caption{$\text{poly}\left( \mathcal{C} \right)$ for $n=3$}
+        \end{figure}
+    \end{minipage}
+\end{frame}
+
+\begin{frame}[t]
+    \frametitle{LP Relaxation}
+   
     \todo{TODO}
 \end{frame}