Finished LP Relaxation slides

latex/presentations/final/presentation.tex

@@ -27,6 +27,7 @@
 \usetikzlibrary{spy}
 \tikzexternalize[prefix=build/]
 
+\captionsetup[sub]{font=small}
 
 \DeclareMathOperator*{\argmax}{arg\,max}
 \DeclareMathOperator*{\argmin}{arg\,min}

latex/presentations/final/sections/decoding_algorithms.tex

@@ -129,118 +129,58 @@ return $\boldsymbol{\hat{c}}$
 \begin{frame}[t]
     \frametitle{LP Decoding \cite{feldman_paper}}
 
-    \begin{minipage}[c]{0.6\linewidth}
     \begin{itemize}
-            \item Codeword polytope:
+        \item General reformulation of ML decoding as a linear program (LP)
-                \begin{align*}
+%        \item Codeword polytope:
-                    \text{poly}\left( \mathcal{C} \right) =
+%            \begin{align*}
-                        \left\{
+%                \text{poly}\left( \mathcal{C} \right) =
-                            \sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}}
+%                    \left\{
-                                \boldsymbol{c} : \alpha_{\boldsymbol{c}} \ge  0,
+%                        \sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}}
-                            \sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}} = 1
+%                            \boldsymbol{c} : \alpha_{\boldsymbol{c}} \ge  0,
-                        \right\},
+%                        \sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}} = 1
-                        \hspace{5mm} \alpha_{\boldsymbol{c}} \in \mathbb{R}_{\ge 0}
+%                    \right\},
-                \end{align*}
+%                    \hspace{5mm} \alpha_{\boldsymbol{c}} \in \mathbb{R}_{\ge 0}
+%            \end{align*}
         \item Cost function:
             \begin{align*}
-                    \boldsymbol{\gamma}^{T} \tilde{\boldsymbol{c}} = \sum_{i=1}^{n}
+                \boldsymbol{\gamma}^{T} \boldsymbol{c} = \sum_{i=1}^{n}
-                        \gamma_i \tilde{c}_i,
+                    \gamma_i c_i,
                 \hspace{5mm}\gamma_i = \ln\left(
                     \frac{p_{Y_i \mid C_i}\left( y_i | c_i = 0 \right) }
                     {p_{Y_i \mid C_i}\left(y_i | c_i=1 \right) } \right)
             \end{align*}
-            \item Exact LP formulation of ML decoding:
+        \item Exact \textit{integer linear programming} (ILP) formulation of ML decoding:
             \begin{align*}
                 &\text{minimize } \boldsymbol{\gamma}^\text{T} \tilde{\boldsymbol{c}}\\
-                    &\text{subject to } \tilde{\boldsymbol{c}}\in\text{poly}
+                &\text{subject to } \boldsymbol{c}\in \mathcal{C}
-                        \left( \mathcal{C} \right) 
             \end{align*}
+        \item Goal: relaxation of constraints to make a practical solution to the problem feasible
     \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) $};
-
-                % f
-
-                \node[color=KITgreen, fill=KITgreen,
-                      draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.7, 0.7, 1) {};
-                \node[color=KITgreen, right=0cm of f] {$\tilde{\boldsymbol{c}}$};
-            \end{tikzpicture}
-            \caption{$\text{poly}\left( \mathcal{C} \right)$ for single parity-check code with
-                $n=3$}
-        \end{figure}
-    \end{minipage}
 \end{frame}
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[t]
-    \frametitle{LP Relaxation}
+    \frametitle{LP Relaxation: Motivation}
 
     \vspace*{-1cm}
 
     \begin{gather*}
-        \boldsymbol{G} =
+%        \boldsymbol{G} =
-            \begin{bmatrix}
+%            \begin{bmatrix}
-                0 & 1 & 1
+%                0 & 1 & 1
-            \end{bmatrix} \hspace{1cm}
+%            \end{bmatrix} \hspace{1cm}
         \boldsymbol{H} =
             \begin{bmatrix}
                 1 & 1 & 1\\
                 0 & 1 & 1
-            \end{bmatrix} \\[1em]
+            \end{bmatrix} \hspace{1cm}
+%            \\[1em]
         \mathcal{C} = \left\{ \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix},
             \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}   \right\} 
     \end{gather*}%
 
-    \vspace*{-1cm}
+    \vspace*{-5mm}
 
     \begin{figure}[H]
         \centering
@@ -356,12 +296,23 @@ return $\boldsymbol{\hat{c}}$
         \end{subfigure}
     \end{figure}
 
+%    \vspace*{-5mm}
+
+    \begin{align*}
+        \text{poly}\left( \mathcal{C} \right) =
+            \left\{
+                \sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}}
+                    \boldsymbol{c} : \alpha_{\boldsymbol{c}} \ge  0,
+                \sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}} = 1
+            \right\},
+            \hspace{5mm} \alpha_{\boldsymbol{c}} \in \mathbb{R}_{\ge 0}
+    \end{align*}
 \end{frame}
 
 \begin{frame}[t]
     \frametitle{LP Relaxation}
   
-    \vspace*{-8mm}
+    \vspace*{-1cm}
 
     \begin{figure}[H]
         \centering
@@ -501,7 +452,7 @@ return $\boldsymbol{\hat{c}}$
                 \end{tikzpicture}
             \end{minipage}
 
-            \caption{Local codeword polytopes of the check nodes}
+            \caption{Local codeword polytopes $\mathcal{P}_{d_j}$ of the check nodes}
             \label{fig:lp:poly:local}
         \end{subfigure}%
         \begin{subfigure}[c]{0.18\textwidth}
@@ -520,6 +471,7 @@ return $\boldsymbol{\hat{c}}$
         \begin{subfigure}[c]{0.32\textwidth}
             \centering
         
+            \vspace*{1.5cm}
             \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
                                 draw, circle, inner sep=0pt, minimum size=4pt]
             \tikzstyle{pseudocodeword} = [color=KITred, fill=KITred,
@@ -575,6 +527,13 @@ return $\boldsymbol{\hat{c}}$
                     {$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
             \end{tikzpicture}
            
+            \vspace*{-5mm}
+
+            \begin{align*}
+                \boldsymbol{T}_j\tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}, \hspace{2mm}
+                    \forall j\in \mathcal{J}
+            \end{align*}
+        
             \caption{Relaxed codeword polytope $\overline{Q}$}
             \label{fig:lp:poly:relaxed}
         \end{subfigure}
@@ -585,10 +544,12 @@ return $\boldsymbol{\hat{c}}$
 \begin{frame}[t]
     \frametitle{LP Relaxation}
   
+    \vspace{-5mm}
+
     \begin{figure}[H]
         \centering
 
-        \begin{subfigure}[c]{0.2\textwidth}
+        \begin{subfigure}[t]{0.3\textwidth}
             \centering
             \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
                                 draw, circle, inner sep=0pt, minimum size=4pt]
@@ -629,20 +590,17 @@ return $\boldsymbol{\hat{c}}$
 
                 \node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
                 \node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
-
-                \node at (3, -2) {$\mathcal{C}$};
             \end{tikzpicture}
-        \end{subfigure}%
-        \begin{subfigure}[c]{0.18\textwidth}
-            \centering
-            \begin{tikzpicture}
-                \node (relaxation) at (0, 0) {Relaxation};
 
-                \draw (-1.5, 0) -- (relaxation);
+            \begin{align*}
-                \draw[->] (relaxation) -- (1.5, 0);
+                \text{minimize}\hspace{2mm} &\boldsymbol{\gamma}^\text{T} \boldsymbol{c} \\
-            \end{tikzpicture}
+                \text{subject to}\hspace{2mm} &\boldsymbol{c} \in \mathcal{C}
+            \end{align*}
+
+            \caption{\textit{Integer linear program} (ILP)}
         \end{subfigure}%
-        \begin{subfigure}[c]{0.2\textwidth}
+        \hfill%
+        \begin{subfigure}[t]{0.3\textwidth}
             \centering
             \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
                                 draw, circle, inner sep=0pt, minimum size=4pt]
@@ -687,20 +645,18 @@ return $\boldsymbol{\hat{c}}$
 
                 \node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
                 \node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
-
-                \node at (3, -2) {$\text{poly}\left(\mathcal{C}\right)$};
             \end{tikzpicture}
-        \end{subfigure}%
-        \begin{subfigure}[c]{0.18\textwidth}
-            \centering
-            \begin{tikzpicture}
-                \node (relaxation) at (0, 0) {Relaxation};
 
-                \draw (-1.5, 0) -- (relaxation);
+            \begin{align*}
-                \draw[->] (relaxation) -- (1.5, 0);
+                \text{minimize}\hspace{2mm} &\boldsymbol{\gamma}^\text{T} \boldsymbol{c} \\
-            \end{tikzpicture}
+                \text{subject to}\hspace{2mm} &\tilde{\boldsymbol{c}} \in
+                    \text{poly}\left( \mathcal{C} \right) &\
+            \end{align*}
+
+            \caption{Motivation}
         \end{subfigure}%
-        \begin{subfigure}[c]{0.2\textwidth}
+        \hfill%
+        \begin{subfigure}[t]{0.3\textwidth}
             \centering
             \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
                                 draw, circle, inner sep=0pt, minimum size=4pt]
@@ -754,12 +710,44 @@ return $\boldsymbol{\hat{c}}$
                 \node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
                 \node[color=KITred, right=0cm of cpseudo]
                     {$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
+            \end{tikzpicture}
             
-                \node at (3, -2) {$\overline{Q}$};
+            \begin{align*}
+                \text{minimize}\hspace{2mm} &\boldsymbol{\gamma}^\text{T} \boldsymbol{c} \\
+                \text{subject to}\hspace{2mm} &
+                    \boldsymbol{T}_j\tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}, \hspace{2mm}
+                        \forall j\in \mathcal{J}
+            \end{align*}
+
+            \caption{\textit{Linear code linear program} (LCLP)}
+        \end{subfigure}%
+    \end{figure}
+
+    \vspace*{-5.75cm}
+    \begin{figure}[H]
+        \centering
+        
+        \begin{subfigure}[t]{0.36\textwidth}
+            \centering
+
+            \begin{tikzpicture}
+                \node (relaxation) at (0, 0) {Relaxation};
+
+                \draw[->] (-1, -0.25) -- (1, -0.25);
+%                \draw (-1.5, 0) -- (relaxation);
+%                \draw[->] (relaxation) -- (1.5, 0);
             \end{tikzpicture}
         \end{subfigure}%
+        \begin{subfigure}[t]{0.31\textwidth}
+            \centering
+            \begin{tikzpicture}
+                \node (relaxation) at (0, 0) {Relaxation};
 
-
+                \draw[->] (-1, -0.25) -- (1, -0.25);
+%                \draw (-1.5, 0) -- (relaxation);
+%                \draw[->] (relaxation) -- (1.5, 0);
+            \end{tikzpicture}
+        \end{subfigure}%
     \end{figure}
 
 \end{frame}