\section{LP Decoding using ADMM}%
\label{sec:LP Decoding using ADMM}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{LP Decoding}%
\label{sub:LP Decoding}

\begin{frame}[t]
	\frametitle{LP Decoding \citereference{FWK05}}

	\vspace*{-3mm}

	\begin{itemize}
		\item General reformulation of ML decoding as a linear program (LP)
		      %        \item Codeword polytope:
		      %            \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*}
		\item Cost function:
		      \begin{align*}
			      \boldsymbol{\gamma}^{T} \boldsymbol{c} = \sum_{i=1}^{n}
			      \gamma_i c_i,
			      \hspace{5mm}\gamma_i = \ln\left(
			      \frac{f_{Y_i \mid C_i}\left( y_i | c_i = 0 \right) }
			      {f_{Y_i \mid C_i}\left(y_i | c_i=1 \right) } \right)
		      \end{align*}
		\item Exact \textit{integer linear programming} (ILP) formulation of ML decoding:
		      \begin{align*}
			       & \text{minimize } \boldsymbol{\gamma}^\text{T} \boldsymbol{c} \\
			       & \text{subject to } \boldsymbol{c}\in \mathcal{C}
		      \end{align*}
		\item Goal: Relaxation of constraints to make a practical solution to the problem feasible
	\end{itemize}

	\bigskip
	\bigskip

	\addreference{FWK05}{J. Feldman; M.J. Wainwright; D.R. Karger: \emph{Using linear programming
			to Decode Binary linear codes}.
		IEEE Transactions on Information Theory 51.3 (2005), pp. 954–972.}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
	\frametitle{LP Decoding: Relaxation}

	\vspace*{-1cm}

	\begin{gather*}
		%        \boldsymbol{G} =
		%            \begin{bmatrix}
		%                0 & 1 & 1
		%            \end{bmatrix} \hspace{1cm}
		\boldsymbol{H} =
		\begin{bmatrix}
			1 & 1 & 1 \\
			0 & 1 & 1
		\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*{-5mm}

	\begin{figure}[H]
		\centering

		\begin{subfigure}[c]{0.4\textwidth}
			\centering

			\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
			draw, circle, inner sep=0pt, minimum size=4pt]

			\tdplotsetmaincoords{60}{25}
			\begin{tikzpicture}[scale=0.9, tdplot_main_coords]
				\tikzstyle{every node}=[font=\normalsize]

				% Cube

				\coordinate (p000) at (0, 0, 0);
				\coordinate (p001) at (0, 0, 2);
				\coordinate (p010) at (0, 2, 0);
				\coordinate (p011) at (0, 2, 2);
				\coordinate (p100) at (2, 0, 0);
				\coordinate (p101) at (2, 0, 2);
				\coordinate (p110) at (2, 2, 0);
				\coordinate (p111) at (2, 2, 2);

				\draw[] (p000) -- (p100);
				\draw[] (p100) -- (p101);
				\draw[] (p101) -- (p001);
				\draw[] (p001) -- (p000);

				\draw[dashed] (p010) -- (p110);
				\draw[]       (p110) -- (p111);
				\draw[]       (p111) -- (p011);
				\draw[dashed] (p011) -- (p010);

				\draw[dashed] (p000) -- (p010);
				\draw[]       (p100) -- (p110);
				\draw[]       (p101) -- (p111);
				\draw[]       (p001) -- (p011);

				% Polytope Vertices

				\node[codeword] (c000) at (p000) {};
				\node[codeword] (c011) at (p011) {};

				% Polytope Annotations

				\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
				\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
			\end{tikzpicture}

			\caption{Set of all codewords $\mathcal{C}$}
			\label{fig:lp:poly:exact_ilp}
		\end{subfigure}%
		\begin{subfigure}[c]{0.16\textwidth}
			\centering
			\begin{tikzpicture}
				\node (relaxation) at (0, 0) {Relaxation};

				\draw (-1.5, 0) -- (relaxation);
				\draw[->] (relaxation) -- (1.5, 0);
			\end{tikzpicture}
		\end{subfigure}%
		\begin{subfigure}[c]{0.4\textwidth}
			\centering

			\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
			draw, circle, inner sep=0pt, minimum size=4pt]

			\tdplotsetmaincoords{60}{25}
			\begin{tikzpicture}[scale=0.9, tdplot_main_coords]
				\tikzstyle{every node}=[font=\normalsize]

				% Cube

				\coordinate (p000) at (0, 0, 0);
				\coordinate (p001) at (0, 0, 2);
				\coordinate (p010) at (0, 2, 0);
				\coordinate (p011) at (0, 2, 2);
				\coordinate (p100) at (2, 0, 0);
				\coordinate (p101) at (2, 0, 2);
				\coordinate (p110) at (2, 2, 0);
				\coordinate (p111) at (2, 2, 2);

				\draw[] (p000) -- (p100);
				\draw[] (p100) -- (p101);
				\draw[] (p101) -- (p001);
				\draw[] (p001) -- (p000);

				\draw[dashed] (p010) -- (p110);
				\draw[]       (p110) -- (p111);
				\draw[]       (p111) -- (p011);
				\draw[dashed] (p011) -- (p010);

				\draw[dashed] (p000) -- (p010);
				\draw[]       (p100) -- (p110);
				\draw[]       (p101) -- (p111);
				\draw[]       (p001) -- (p011);

				% Polytope Vertices

				\node[codeword] (c000) at (p000) {};
				\node[codeword] (c011) at (p011) {};

				% Polytope Edges

				\draw[line width=1pt, color=KITblue] (c000) -- (c011);

				% Polytope Annotations

				\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
				\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
			\end{tikzpicture}

			\caption{Codeword polytope $\text{poly}\left( \mathcal{C} \right) $}
			\label{fig:lp:poly:exact}
		\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 Decoding: Relaxation}

	\vspace*{-1cm}

	\begin{figure}[H]
		\centering

		\begin{subfigure}[c]{0.48\textwidth}
			\centering

			\begin{minipage}{\textwidth}
				\centering

				\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
				draw, circle, inner sep=0pt, minimum size=4pt]

				\tdplotsetmaincoords{60}{25}
				\begin{tikzpicture}[scale=0.7, tdplot_main_coords]
					\tikzstyle{every node}=[font=\normalsize]

					% Cube

					\coordinate (p000) at (0, 0, 0);
					\coordinate (p001) at (0, 0, 2);
					\coordinate (p010) at (0, 2, 0);
					\coordinate (p011) at (0, 2, 2);
					\coordinate (p100) at (2, 0, 0);
					\coordinate (p101) at (2, 0, 2);
					\coordinate (p110) at (2, 2, 0);
					\coordinate (p111) at (2, 2, 2);

					\draw[] (p000) -- (p100);
					\draw[] (p100) -- (p101);
					\draw[] (p101) -- (p001);
					\draw[] (p001) -- (p000);

					\draw[dashed] (p010) -- (p110);
					\draw[]       (p110) -- (p111);
					\draw[]       (p111) -- (p011);
					\draw[dashed] (p011) -- (p010);

					\draw[dashed] (p000) -- (p010);
					\draw[]       (p100) -- (p110);
					\draw[]       (p101) -- (p111);
					\draw[]       (p001) -- (p011);

					% Polytope Vertices

					\node[codeword] (c000) at (p000) {};
					\node[codeword] (c101) at (p101) {};
					\node[codeword] (c110) at (p110) {};
					\node[codeword] (c011) at (p011) {};

					% Polytope Edges & Faces

					\draw[line width=1pt, color=KITblue] (c000) -- (c101);
					\draw[line width=1pt, color=KITblue] (c000) -- (c110);
					\draw[line width=1pt, color=KITblue] (c000) -- (c011);

					\draw[line width=1pt, color=KITblue] (c101) -- (c110);
					\draw[line width=1pt, color=KITblue] (c101) -- (c011);

					\draw[line width=1pt, color=KITblue] (c011) -- (c110);

					\fill[KITblue, opacity=0.15] (p000) -- (p101) -- (p011) -- cycle;
					\fill[KITblue, opacity=0.15] (p000) -- (p110) -- (p101) -- cycle;
					\fill[KITblue, opacity=0.15] (p110) -- (p011) -- (p101) -- cycle;

					% Polytope Annotations

					\node[color=KITblue, below=0cm of c000]    {$\left( 0, 0, 0 \right) $};
					\node[color=KITblue, right=0.07cm of c101] {$\left( 1, 0, 1 \right) $};
					\node[color=KITblue, right=0cm of c110]    {$\left( 1, 1, 0 \right) $};
					\node[color=KITblue, above=0cm of c011]    {$\left( 0, 1, 1 \right) $};

					\node[color=KITblue, align=center] at (-4,1)
					{$j=1$\\ $\left( c_1 + c_2+ c_3 = 0 \right) $};
				\end{tikzpicture}
			\end{minipage}

			\vspace{5mm}

			\begin{minipage}{\textwidth}
				\centering

				\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
				draw, circle, inner sep=0pt, minimum size=4pt]

				\tdplotsetmaincoords{60}{25}
				\begin{tikzpicture}[scale=0.7, tdplot_main_coords]
					\tikzstyle{every node}=[font=\normalsize]

					% Cube

					\coordinate (p000) at (0, 0, 0);
					\coordinate (p001) at (0, 0, 2);
					\coordinate (p010) at (0, 2, 0);
					\coordinate (p011) at (0, 2, 2);
					\coordinate (p100) at (2, 0, 0);
					\coordinate (p101) at (2, 0, 2);
					\coordinate (p110) at (2, 2, 0);
					\coordinate (p111) at (2, 2, 2);

					\draw[] (p000) -- (p100);
					\draw[] (p100) -- (p101);
					\draw[] (p101) -- (p001);
					\draw[] (p001) -- (p000);

					\draw[dashed] (p010) -- (p110);
					\draw[]       (p110) -- (p111);
					\draw[]       (p111) -- (p011);
					\draw[dashed] (p011) -- (p010);

					\draw[dashed] (p000) -- (p010);
					\draw[]       (p100) -- (p110);
					\draw[]       (p101) -- (p111);
					\draw[]       (p001) -- (p011);

					% Polytope Vertices

					\node[codeword] (c000) at (p000) {};
					\node[codeword] (c011) at (p011) {};
					\node[codeword] (c100) at (p100) {};
					\node[codeword] (c111) at (p111) {};

					% Polytope Edges & Faces

					\draw[line width=1pt, color=KITblue] (c000) -- (c011);
					\draw[line width=1pt, color=KITblue] (c000) -- (c100);
					\draw[line width=1pt, color=KITblue] (c100) -- (c111);
					\draw[line width=1pt, color=KITblue] (c111) -- (c011);

					\fill[KITblue, opacity=0.2] (p000) -- (p100) -- (p111) -- (p011) -- cycle;

					% Polytope Annotations

					\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
					\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
					\node[color=KITblue, below=0cm of c100] {$\left( 1, 0, 0 \right) $};
					\node[color=KITblue, above=0cm of c111] {$\left( 1, 1, 1 \right) $};

					\node[color=KITblue, align=center] at (-4,1)
					{$j=2$\\ $\left(c_2 + c_3 = 0\right)$};
				\end{tikzpicture}
			\end{minipage}

			\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}
			\centering
			\begin{tikzpicture}
				\draw[densely dashed] (0, -2) -- (0, 2);
				\draw[densely dashed] (-0.5, -2) -- (0, -2);
				\draw[densely dashed] (-0.5, 2) -- (0, 2);

				\node (intersection) at (1.5, 0) {Intersection};

				\draw[densely dashed] (0, 0) -- (intersection);
				\draw[densely dashed, ->] (intersection) -- (3,0);
			\end{tikzpicture}
		\end{subfigure}%
		\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,
			draw, circle, inner sep=0pt, minimum size=4pt]

			\tdplotsetmaincoords{60}{25}
			\begin{tikzpicture}[scale=0.9, tdplot_main_coords]
				\tikzstyle{every node}=[font=\normalsize]

				% Cube

				\coordinate (p000) at (0, 0, 0);
				\coordinate (p001) at (0, 0, 2);
				\coordinate (p010) at (0, 2, 0);
				\coordinate (p011) at (0, 2, 2);
				\coordinate (p100) at (2, 0, 0);
				\coordinate (p101) at (2, 0, 2);
				\coordinate (p110) at (2, 2, 0);
				\coordinate (p111) at (2, 2, 2);

				\draw[] (p000) -- (p100);
				\draw[] (p100) -- (p101);
				\draw[] (p101) -- (p001);
				\draw[] (p001) -- (p000);

				\draw[dashed] (p010) -- (p110);
				\draw[]       (p110) -- (p111);
				\draw[]       (p111) -- (p011);
				\draw[dashed] (p011) -- (p010);

				\draw[dashed] (p000) -- (p010);
				\draw[]       (p100) -- (p110);
				\draw[]       (p101) -- (p111);
				\draw[]       (p001) -- (p011);

				% Polytope Vertices

				\node[codeword] (c000) at (p000) {};
				\node[codeword] (c011) at (p011) {};
				\node[pseudocodeword] (cpseudo) at (2, 1, 1) {};

				% Polytope Edges & Faces

				\draw[line width=1pt, color=KITblue] (c000) -- (c011);
				\draw[line width=1pt, color=KITred] (cpseudo) -- (c000);
				\draw[line width=1pt, color=KITred] (cpseudo) -- (c011);

				\fill[KITred, opacity=0.2] (p000) -- (p011) -- (2,1,1) -- cycle;

				% Polytope Annotations

				\node[color=KITblue, below=0cm of c000]   {$\left( 0, 0, 0 \right) $};
				\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
				\node[color=KITred, right=0cm of cpseudo]
				{$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
			\end{tikzpicture}

			\vspace*{-5mm}

			\begin{gather*}
				\boldsymbol{T}_j\tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}, \hspace{2mm}
				\forall j\in \mathcal{J} \\[0.5em]
				\boldsymbol{T}_j \in \mathbb{F}_2^{d_j \times n}
			\end{gather*}

			\vspace*{-2mm}

			\caption{Relaxed codeword polytope $\overline{Q}$}
			\label{fig:lp:poly:relaxed}
		\end{subfigure}
	\end{figure}
\end{frame}

\begin{frame}[t]
	\frametitle{LP Decoding: Relaxation}

	\vspace{-5mm}

	\begin{figure}[H]
		\centering

		\begin{subfigure}[t]{0.3\textwidth}
			\centering
			\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
			draw, circle, inner sep=0pt, minimum size=4pt]
			\tdplotsetmaincoords{60}{25}
			\begin{tikzpicture}[scale=0.8, tdplot_main_coords]
				\tikzstyle{every node}=[font=\normalsize]

				% Cube

				\coordinate (p000) at (0, 0, 0);
				\coordinate (p001) at (0, 0, 2);
				\coordinate (p010) at (0, 2, 0);
				\coordinate (p011) at (0, 2, 2);
				\coordinate (p100) at (2, 0, 0);
				\coordinate (p101) at (2, 0, 2);
				\coordinate (p110) at (2, 2, 0);
				\coordinate (p111) at (2, 2, 2);

				\draw[] (p000) -- (p100);
				\draw[] (p100) -- (p101);
				\draw[] (p101) -- (p001);
				\draw[] (p001) -- (p000);

				\draw[dashed] (p010) -- (p110);
				\draw[]       (p110) -- (p111);
				\draw[]       (p111) -- (p011);
				\draw[dashed] (p011) -- (p010);

				\draw[dashed] (p000) -- (p010);
				\draw[]       (p100) -- (p110);
				\draw[]       (p101) -- (p111);
				\draw[]       (p001) -- (p011);

				% Polytope Vertices

				\node[codeword] (c000) at (p000) {};
				\node[codeword] (c011) at (p011) {};

				% Polytope Annotations

				\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
				\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
			\end{tikzpicture}

			\begin{align*}
				\text{minimize}\hspace{2mm}   & \boldsymbol{\gamma}^\text{T} \boldsymbol{c} \\
				\text{subject to}\hspace{2mm} & \boldsymbol{c} \in \mathcal{C}
			\end{align*}

			\caption{\textit{Integer linear program} (ILP)}
		\end{subfigure}%
		\hfill%
		\begin{subfigure}[t]{0.3\textwidth}
			\centering
			\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
			draw, circle, inner sep=0pt, minimum size=4pt]
			\tdplotsetmaincoords{60}{25}
			\begin{tikzpicture}[scale=0.8, tdplot_main_coords]
				\tikzstyle{every node}=[font=\normalsize]

				% Cube

				\coordinate (p000) at (0, 0, 0);
				\coordinate (p001) at (0, 0, 2);
				\coordinate (p010) at (0, 2, 0);
				\coordinate (p011) at (0, 2, 2);
				\coordinate (p100) at (2, 0, 0);
				\coordinate (p101) at (2, 0, 2);
				\coordinate (p110) at (2, 2, 0);
				\coordinate (p111) at (2, 2, 2);

				\draw[] (p000) -- (p100);
				\draw[] (p100) -- (p101);
				\draw[] (p101) -- (p001);
				\draw[] (p001) -- (p000);

				\draw[dashed] (p010) -- (p110);
				\draw[]       (p110) -- (p111);
				\draw[]       (p111) -- (p011);
				\draw[dashed] (p011) -- (p010);

				\draw[dashed] (p000) -- (p010);
				\draw[]       (p100) -- (p110);
				\draw[]       (p101) -- (p111);
				\draw[]       (p001) -- (p011);

				% Polytope Vertices

				\node[codeword] (c000) at (p000) {};
				\node[codeword] (c011) at (p011) {};

				% Polytope Edges

				\draw[line width=1pt, color=KITblue] (c000) -- (c011);

				% Polytope Annotations

				\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
				\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
			\end{tikzpicture}

			\begin{align*}
				\text{minimize}\hspace{2mm}           & \boldsymbol{\gamma}^\text{T} \tilde{\boldsymbol{c}} \\
				\text{subject to}\hspace{2mm}         & \tilde{\boldsymbol{c}} \in
				\text{poly}\left( \mathcal{C} \right) & \
			\end{align*}

			\caption{Motivation}
		\end{subfigure}%
		\hfill%
		\begin{subfigure}[t]{0.3\textwidth}
			\centering
			\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
			draw, circle, inner sep=0pt, minimum size=4pt]
			\tikzstyle{pseudocodeword} = [color=KITred, fill=KITred,
			draw, circle, inner sep=0pt, minimum size=4pt]
			\tdplotsetmaincoords{60}{25}
			\begin{tikzpicture}[scale=0.8, tdplot_main_coords]
				\tikzstyle{every node}=[font=\normalsize]

				% Cube

				\coordinate (p000) at (0, 0, 0);
				\coordinate (p001) at (0, 0, 2);
				\coordinate (p010) at (0, 2, 0);
				\coordinate (p011) at (0, 2, 2);
				\coordinate (p100) at (2, 0, 0);
				\coordinate (p101) at (2, 0, 2);
				\coordinate (p110) at (2, 2, 0);
				\coordinate (p111) at (2, 2, 2);

				\draw[] (p000) -- (p100);
				\draw[] (p100) -- (p101);
				\draw[] (p101) -- (p001);
				\draw[] (p001) -- (p000);

				\draw[dashed] (p010) -- (p110);
				\draw[]       (p110) -- (p111);
				\draw[]       (p111) -- (p011);
				\draw[dashed] (p011) -- (p010);

				\draw[dashed] (p000) -- (p010);
				\draw[]       (p100) -- (p110);
				\draw[]       (p101) -- (p111);
				\draw[]       (p001) -- (p011);

				% Polytope Vertices

				\node[codeword] (c000) at (p000) {};
				\node[codeword] (c011) at (p011) {};
				\node[pseudocodeword] (cpseudo) at (2, 1, 1) {};

				% Polytope Edges & Faces

				\draw[line width=1pt, color=KITblue] (c000) -- (c011);
				\draw[line width=1pt, color=KITred] (cpseudo) -- (c000);
				\draw[line width=1pt, color=KITred] (cpseudo) -- (c011);

				\fill[KITred, opacity=0.2] (p000) -- (p011) -- (2,1,1) -- cycle;

				% Polytope Annotations

				\node[color=KITblue, below=0cm of c000]   {$\left( 0, 0, 0 \right) $};
				\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
				\node[color=KITred, right=0cm of cpseudo]
				{$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
			\end{tikzpicture}

			\begin{align*}
				\text{minimize}\hspace{2mm}   & \boldsymbol{\gamma}^\text{T} \tilde{\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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
	\frametitle{LP Decoding using ADMM}

	\vspace*{-4mm}

	\begin{itemize}
		\item Solution using the \textit{alternating direction method of multipliers} (ADMM)
		      \citereference{$\text{Bar}^+\text{13}$}
		\item Slight reformulation of the LCLP:
		      \begin{align*}
			      \begin{aligned}
				      \text{minimize}\hspace{2mm}   & \boldsymbol{\gamma}^\text{T}
				      \tilde{\boldsymbol{c}}
				      + \sum_{j\in\mathcal{J}} g_j\left( \boldsymbol{z}_j \right)  \\
				      \text{subject to}\hspace{2mm} &
				      \boldsymbol{T}_j\tilde{\boldsymbol{c}} = \boldsymbol{z}_j, \hspace{2mm}
				      \forall j\in \mathcal{J}
			      \end{aligned}\hspace{2mm},\hspace{1cm}
			      g_j\left( \boldsymbol{t} \right)  := \begin{cases}
				                                           0,       & \boldsymbol{t} \in \mathcal{P}_{d_j}     \\
				                                           +\infty, & \boldsymbol{t} \not\in \mathcal{P}_{d_j}
			                                           \end{cases}
		      \end{align*}
		\item Iterative algorithm:
		      \begin{alignat*}{3}
			      \tilde{\boldsymbol{c}} & \leftarrow \argmin_{\tilde{\boldsymbol{c}}}
			      \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
			      + \frac{\mu}{2}\sum_{j\in\mathcal{J}} \left\Vert
			      \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j
			      + \boldsymbol{u}_j \right\Vert                                                                 \\
			      \boldsymbol{z}_j       & \leftarrow \argmin_{\boldsymbol{z}_j}
			      g\left( \boldsymbol{z}_j \right)
			      + \frac{\mu}{2} \left\Vert \boldsymbol{T}_j \tilde{\boldsymbol{c}}
			      - \boldsymbol{z}_j + \boldsymbol{u}_j \right\Vert,
			      \hspace{5mm}           &                                             & \forall j\in\mathcal{J} \\
			      \boldsymbol{u}_j       & \leftarrow \boldsymbol{u}_j
			      + \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j,
			      \hspace{5mm}           &                                             & \forall j\in\mathcal{J}
			      %                    \left( g\left( \boldsymbol{\boldsymbol{z}_j} \right)
			      %                    + \frac{\mu}{2} \left\Vert \boldsymbol{T}_j\tilde{\boldsymbol{c}}
			      %                    - \boldsymbol{z}_j + \boldsymbol{u}_j\right\Vert \right) 
		      \end{alignat*}
	\end{itemize}

	\bigskip

	\addreference{$\text{Bar}^+\text{13}$}{Siddharth Barman et al.:
		\emph{Decomposition Methods for Large Scale LP Decoding}.
		IEEE Transactions on Information Theory 59.12 (2013), pp. 7870–7886.}
\end{frame}

%%%%%%%%%%%%%%%%
\subsection{Application of ADMM}
\label{subsec:Application of ADMM}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
	\frametitle{LP Decoding using ADMM}

	\vspace*{-7mm}

	\begin{itemize}
		\item Convergence properties enhanced by over-relaxation with parameter $\rho$
		\item Simplified rules:
		      %            \footnote{$\left( \boldsymbol{z}_j \right)_i $ is a slight abuse of notation.
		      %            What is actually meant is the component of $\boldsymbol{z}_j$ that is associated
		      %            with the VN $i$, i.e., $\left( \boldsymbol{T}_j^\text{T}
		      %                \boldsymbol{z}_j \right)_i$.\\
		      %            The same is true for $\left( \boldsymbol{u}_j \right)_i$}:

		      \vspace*{-4mm}

		      \begin{alignat*}{3}
			      \tilde{c}_i      & \leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left(
			      \sum_{j\in N_v\left( i \right) } \Big( \left( \boldsymbol{z}_j \right)_i
				      - \left( \boldsymbol{u}_j \right)_i \Big)
			      - \frac{\gamma_i}{\mu} \right)
			      \hspace{5mm}     &                                                                & \forall i\in\mathcal{I} \\
			      \boldsymbol{z}_j & \leftarrow \Pi_{\mathcal{P}_{d_j}}\left(
			      \rho \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j
			      \left( 1-\rho \right) + \boldsymbol{u}_j \right)
			      \hspace{5mm}     &                                                                & \forall j\in\mathcal{J} \\
			      \boldsymbol{u}_j & \leftarrow \boldsymbol{u}_j
			      + \rho \boldsymbol{T}_j\tilde{\boldsymbol{c}}
			      - \left( 1-\rho \right) \boldsymbol{z}_j
			      - \boldsymbol{z}_j
			      \hspace{5mm}     &                                                                & \forall j\in\mathcal{J}
		      \end{alignat*}

		      \vspace*{2mm}

		\item The projections $\Pi_{\mathcal{P}_{d_j}}, \hspace{1mm} j\in\mathcal{J}$
		      are the main computational effort
		\item Many approaches exist \citereference{$\text{Bar}^+\text{13}$},
		      \citereference{ZS13}, \citereference{$\text{Gen}^+\text{20}$}
		\item The approach chosen here is the one described in
		      \citereference{$\text{Bar}^+\text{13}$}
	\end{itemize}

	\bigskip
	\bigskip

	\addreferences
	{$\text{Bar}^+\text{13}$}{Siddharth Barman et al.:
		\emph{Decomposition Methods for Large Scale LP Decoding}.
		IEEE Transactions on Information Theory 59.12 (2013), pp. 7870–7886.}
	{ZS13}{Xiaojie Zhang; Paul H. Siegel: \emph{Efficient iterative LP decoding
			of LDPC codes with alternating direction method of multipliers}.
		2013 IEEE International Symposium on Information Theory. 2013, pp. 1501–1505.}
	{$\text{Gen}^+\text{20}$}{Florian Gensheimer et al.:
		\emph{A Reduced-Complexity Projection Algorithm for ADMM-Based LP Decoding}.
		IEEE Transactions on Information Theory 66.8 (2020), pp. 4819–4833.}
	\stopreferences
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simulation Results}%
\label{sub:Simulation Results}

\begin{frame}[t]
	\frametitle{LP Decoding using ADMM: Frame Error Rate}

	\vspace*{-6mm}

	\begin{itemize}
        \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac24, 204.33.484}
	\end{itemize}

	\bigskip

	\begin{figure}[H]
		\centering

		\begin{tikzpicture}
			\begin{axis}[
					grid=both,
					xlabel={$E_b / N_0 \left( \text{dB} \right) $}, ylabel={FER},
					ymode=log,
					width=0.45\textwidth,
					height=0.325\textwidth,
					%legend style={at={(0.03,0.04)},anchor=south west},
					legend pos=outer north east,
					legend cell align={left},
				]

                \addplot[scol1, line width=1pt]
                    table [x=SNR, y=FER, col sep=comma,
                           discard if not={mu}{3.0},
                           discard if gt={SNR}{5.5}]
                        {res/admm/ber_2d_20433484.csv};
                \addlegendentry{ADMM}
                \addplot [KITorange, line width=1pt]
                    table [x=SNR, y=FER, col sep=comma,
                           discard if gt={SNR}{5.5}]
                        {res/generic/bp_20433484.csv};

                \addlegendentry{BP}
                \addplot[black, line width=1pt]
                    table [col sep=comma, x=SNR, y=FER,
                           discard if gt={SNR}{5.5}]
                        {res/generic/fer_ml_20433484.csv};
                \addlegendentry{ML \citereference{HEL+19}}
			\end{axis}
		\end{tikzpicture}
	\end{figure}%

	\bigskip
	\bigskip

	\addreferences
	{Mac24}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
		June 2024. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
    {Hel+19}{Michael Helmling, Stefan Scholl, Florian Gensheimer, Tobias Dietz, Kira Kraft,
        Stefan Ruzika, and Norbert Wehn. \emph{Database of Channel Codes and 
        ML Simulation Results}. June 2024. URL: \url{www.rptu.de/channel-codes}}
	\stopreferences
\end{frame}