Finished LP Relaxation slides
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@ -27,6 +27,7 @@
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\usetikzlibrary{spy}
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\tikzexternalize[prefix=build/]
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\captionsetup[sub]{font=small}
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\DeclareMathOperator*{\argmax}{arg\,max}
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\DeclareMathOperator*{\argmin}{arg\,min}
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@ -129,118 +129,58 @@ return $\boldsymbol{\hat{c}}$
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\begin{frame}[t]
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\frametitle{LP Decoding \cite{feldman_paper}}
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\begin{minipage}[c]{0.6\linewidth}
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\begin{itemize}
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\item Codeword polytope:
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\begin{align*}
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\text{poly}\left( \mathcal{C} \right) =
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\left\{
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\sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}}
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\boldsymbol{c} : \alpha_{\boldsymbol{c}} \ge 0,
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\sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}} = 1
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\right\},
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\hspace{5mm} \alpha_{\boldsymbol{c}} \in \mathbb{R}_{\ge 0}
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\end{align*}
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\item Cost function:
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\begin{align*}
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\boldsymbol{\gamma}^{T} \tilde{\boldsymbol{c}} = \sum_{i=1}^{n}
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\gamma_i \tilde{c}_i,
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\hspace{5mm}\gamma_i = \ln\left(
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\frac{p_{Y_i \mid C_i}\left( y_i | c_i = 0 \right) }
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{p_{Y_i \mid C_i}\left(y_i | c_i=1 \right) } \right)
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\end{align*}
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\item Exact LP formulation of ML decoding:
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\begin{align*}
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&\text{minimize } \boldsymbol{\gamma}^\text{T} \tilde{\boldsymbol{c}}\\
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&\text{subject to } \tilde{\boldsymbol{c}}\in\text{poly}
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\left( \mathcal{C} \right)
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\end{align*}
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\end{itemize}
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\end{minipage}%
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\hfill%
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\begin{minipage}[c]{0.4\linewidth}
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\begin{figure}[H]
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{245}
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\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
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% Cube
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\draw[dashed] (0, 0, 0) -- (2, 0, 0);
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\draw[dashed] (2, 0, 0) -- (2, 0, 2);
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\draw[] (2, 0, 2) -- (0, 0, 2);
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\draw[] (0, 0, 2) -- (0, 0, 0);
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\draw[] (0, 2, 0) -- (2, 2, 0);
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\draw[] (2, 2, 0) -- (2, 2, 2);
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\draw[] (2, 2, 2) -- (0, 2, 2);
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\draw[] (0, 2, 2) -- (0, 2, 0);
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\draw[] (0, 0, 0) -- (0, 2, 0);
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\draw[dashed] (2, 0, 0) -- (2, 2, 0);
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\draw[] (2, 0, 2) -- (2, 2, 2);
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\draw[] (0, 0, 2) -- (0, 2, 2);
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% Codeword Polytope
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\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 0, 2);
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\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 2, 0);
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\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (0, 2, 2);
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\draw[line width=1pt, color=KITblue] (2, 0, 2) -- (2, 2, 0);
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\draw[line width=1pt, color=KITblue] (2, 0, 2) -- (0, 2, 2);
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\draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);
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% Polytope Annotations
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\node[codeword] (c000) at (0, 0, 0) {};% {$\left( 0, 0, 0 \right) $};
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\node[codeword] (c101) at (2, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
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\node[codeword] (c110) at (2, 2, 0) {};% {$\left( 1, 1, 0 \right) $};
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\node[codeword] (c011) at (0, 2, 2) {};% {$\left( 0, 1, 1 \right) $};
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\node[color=KITblue, right=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c101] {$\left( 1, 0, 1 \right) $};
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\node[color=KITblue, left=0cm of c110] {$\left( 1, 1, 0 \right) $};
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\node[color=KITblue, left=0cm of c011] {$\left( 0, 1, 1 \right) $};
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% f
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\node[color=KITgreen, fill=KITgreen,
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draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.7, 0.7, 1) {};
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\node[color=KITgreen, right=0cm of f] {$\tilde{\boldsymbol{c}}$};
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\end{tikzpicture}
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\caption{$\text{poly}\left( \mathcal{C} \right)$ for single parity-check code with
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$n=3$}
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\end{figure}
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\end{minipage}
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\begin{itemize}
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\item General reformulation of ML decoding as a linear program (LP)
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% \item Codeword polytope:
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% \begin{align*}
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% \text{poly}\left( \mathcal{C} \right) =
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% \left\{
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% \sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}}
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% \boldsymbol{c} : \alpha_{\boldsymbol{c}} \ge 0,
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% \sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}} = 1
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% \right\},
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% \hspace{5mm} \alpha_{\boldsymbol{c}} \in \mathbb{R}_{\ge 0}
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% \end{align*}
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\item Cost function:
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\begin{align*}
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\boldsymbol{\gamma}^{T} \boldsymbol{c} = \sum_{i=1}^{n}
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\gamma_i c_i,
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\hspace{5mm}\gamma_i = \ln\left(
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\frac{p_{Y_i \mid C_i}\left( y_i | c_i = 0 \right) }
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{p_{Y_i \mid C_i}\left(y_i | c_i=1 \right) } \right)
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\end{align*}
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\item Exact \textit{integer linear programming} (ILP) formulation of ML decoding:
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\begin{align*}
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&\text{minimize } \boldsymbol{\gamma}^\text{T} \tilde{\boldsymbol{c}}\\
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&\text{subject to } \boldsymbol{c}\in \mathcal{C}
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\end{align*}
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\item Goal: relaxation of constraints to make a practical solution to the problem feasible
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{LP Relaxation}
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\frametitle{LP Relaxation: Motivation}
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\vspace*{-1cm}
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\begin{gather*}
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\boldsymbol{G} =
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\begin{bmatrix}
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0 & 1 & 1
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\end{bmatrix} \hspace{1cm}
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% \boldsymbol{G} =
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% \begin{bmatrix}
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% 0 & 1 & 1
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% \end{bmatrix} \hspace{1cm}
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\boldsymbol{H} =
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\begin{bmatrix}
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1 & 1 & 1\\
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0 & 1 & 1
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\end{bmatrix} \\[1em]
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\end{bmatrix} \hspace{1cm}
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% \\[1em]
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\mathcal{C} = \left\{ \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix},
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\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\}
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\end{gather*}%
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\vspace*{-1cm}
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\vspace*{-5mm}
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\begin{figure}[H]
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\centering
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@ -356,12 +296,23 @@ return $\boldsymbol{\hat{c}}$
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\end{subfigure}
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\end{figure}
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% \vspace*{-5mm}
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\begin{align*}
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\text{poly}\left( \mathcal{C} \right) =
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\left\{
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\sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}}
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\boldsymbol{c} : \alpha_{\boldsymbol{c}} \ge 0,
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\sum_{\boldsymbol{c}\in\mathcal{C}}\alpha_{\boldsymbol{c}} = 1
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\right\},
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\hspace{5mm} \alpha_{\boldsymbol{c}} \in \mathbb{R}_{\ge 0}
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\end{align*}
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\end{frame}
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\begin{frame}[t]
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\frametitle{LP Relaxation}
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\vspace*{-8mm}
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\vspace*{-1cm}
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\begin{figure}[H]
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\centering
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@ -501,7 +452,7 @@ return $\boldsymbol{\hat{c}}$
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\end{tikzpicture}
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\end{minipage}
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\caption{Local codeword polytopes of the check nodes}
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\caption{Local codeword polytopes $\mathcal{P}_{d_j}$ of the check nodes}
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\label{fig:lp:poly:local}
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\end{subfigure}%
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\begin{subfigure}[c]{0.18\textwidth}
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@ -520,6 +471,7 @@ return $\boldsymbol{\hat{c}}$
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\begin{subfigure}[c]{0.32\textwidth}
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\centering
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\vspace*{1.5cm}
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tikzstyle{pseudocodeword} = [color=KITred, fill=KITred,
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@ -575,6 +527,13 @@ return $\boldsymbol{\hat{c}}$
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{$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
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\end{tikzpicture}
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\vspace*{-5mm}
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\begin{align*}
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\boldsymbol{T}_j\tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}, \hspace{2mm}
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\forall j\in \mathcal{J}
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\end{align*}
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\caption{Relaxed codeword polytope $\overline{Q}$}
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\label{fig:lp:poly:relaxed}
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\end{subfigure}
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@ -585,10 +544,12 @@ return $\boldsymbol{\hat{c}}$
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\begin{frame}[t]
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\frametitle{LP Relaxation}
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\vspace{-5mm}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[c]{0.2\textwidth}
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\begin{subfigure}[t]{0.3\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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@ -629,20 +590,17 @@ return $\boldsymbol{\hat{c}}$
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\node at (3, -2) {$\mathcal{C}$};
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}[c]{0.18\textwidth}
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\centering
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\begin{tikzpicture}
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\node (relaxation) at (0, 0) {Relaxation};
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\draw (-1.5, 0) -- (relaxation);
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\draw[->] (relaxation) -- (1.5, 0);
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\end{tikzpicture}
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\begin{align*}
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\text{minimize}\hspace{2mm} &\boldsymbol{\gamma}^\text{T} \boldsymbol{c} \\
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\text{subject to}\hspace{2mm} &\boldsymbol{c} \in \mathcal{C}
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\end{align*}
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\caption{\textit{Integer linear program} (ILP)}
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\end{subfigure}%
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\begin{subfigure}[c]{0.2\textwidth}
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\hfill%
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\begin{subfigure}[t]{0.3\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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@ -687,20 +645,18 @@ return $\boldsymbol{\hat{c}}$
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\node at (3, -2) {$\text{poly}\left(\mathcal{C}\right)$};
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}[c]{0.18\textwidth}
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\centering
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\begin{tikzpicture}
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\node (relaxation) at (0, 0) {Relaxation};
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\draw (-1.5, 0) -- (relaxation);
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\draw[->] (relaxation) -- (1.5, 0);
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\end{tikzpicture}
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\begin{align*}
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\text{minimize}\hspace{2mm} &\boldsymbol{\gamma}^\text{T} \boldsymbol{c} \\
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\text{subject to}\hspace{2mm} &\tilde{\boldsymbol{c}} \in
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\text{poly}\left( \mathcal{C} \right) &\
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\end{align*}
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\caption{Motivation}
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\end{subfigure}%
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\begin{subfigure}[c]{0.2\textwidth}
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\hfill%
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\begin{subfigure}[t]{0.3\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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@ -754,12 +710,44 @@ return $\boldsymbol{\hat{c}}$
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\node[color=KITred, right=0cm of cpseudo]
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{$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
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\end{tikzpicture}
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\node at (3, -2) {$\overline{Q}$};
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\begin{align*}
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\text{minimize}\hspace{2mm} &\boldsymbol{\gamma}^\text{T} \boldsymbol{c} \\
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\text{subject to}\hspace{2mm} &
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\boldsymbol{T}_j\tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}, \hspace{2mm}
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\forall j\in \mathcal{J}
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\end{align*}
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\caption{\textit{Linear code linear program} (LCLP)}
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\end{subfigure}%
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\end{figure}
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\vspace*{-5.75cm}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[t]{0.36\textwidth}
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\centering
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\begin{tikzpicture}
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\node (relaxation) at (0, 0) {Relaxation};
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\draw[->] (-1, -0.25) -- (1, -0.25);
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% \draw (-1.5, 0) -- (relaxation);
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% \draw[->] (relaxation) -- (1.5, 0);
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}[t]{0.31\textwidth}
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\centering
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\begin{tikzpicture}
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\node (relaxation) at (0, 0) {Relaxation};
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\draw[->] (-1, -0.25) -- (1, -0.25);
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% \draw (-1.5, 0) -- (relaxation);
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% \draw[->] (relaxation) -- (1.5, 0);
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\end{tikzpicture}
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\end{subfigure}%
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\end{figure}
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\end{frame}
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