Moved LP Decoding slide; Added general optimization slide; Added forthcoming examination slide
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@ -77,3 +77,25 @@
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institution = {KIT},
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}
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@BOOK{distr_opt_book,
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author = {Boyd, Stephen and Parikh, Neal and Chu, Eric and Peleato, Borja and Eckstein, Jonathan},
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booktitle = {Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers},
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year = {2011},
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volume = {},
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number = {},
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pages = {},
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doi = {},
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url = {https://ieeexplore.ieee.org/document/8186925},
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}
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@INPROCEEDINGS{efficient_lp_dec_admm,
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author = {Zhang, Xiaojie and Siegel, Paul H.},
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booktitle = {2013 IEEE International Symposium on Information Theory},
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title = {Efficient iterative LP decoding of LDPC codes with alternating direction method of multipliers},
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year = {2013},
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volume = {},
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number = {},
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pages = {1501-1505},
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doi = {10.1109/ISIT.2013.6620477}
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}
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@ -46,6 +46,8 @@
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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{Proximal Decoding: General Idea}
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@ -82,6 +84,8 @@
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t, fragile]
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\frametitle{Proximal Decoding: Algorithm}
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\begin{itemize}
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@ -104,18 +108,199 @@ Output $\boldsymbol{\hat{x}}$
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{ADMM}%
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\label{sub:Alg ADMM}
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\subsection{LP Decoding}%
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\label{sub:LP Decoding}
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\begin{frame}[t]
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\frametitle{ADMM Decoding: General Idea}
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\frametitle{LP Decoding}
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\todo{TODO}
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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}}\lambda_{\boldsymbol{c}}
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\boldsymbol{c} : \lambda_{\boldsymbol{c}} \ge 0,
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\sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}} = 1
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\right\},
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\hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R}
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\end{align*}
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\item Cost Function:
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\begin{align*}
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\sum_{i=1}^{n} \gamma_i c_i,
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\hspace{5mm}\gamma_i = \log\left(
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\frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
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\end{align*}
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\item LP Formulation of ML Decoding:
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\begin{align*}
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&\text{minimize } \sum_{i=1}^{n} \gamma_i f_i \\
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&\text{subject to } \boldsymbol{f}\in\text{poly}\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) $};
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% f
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\node[color=KITgreen, fill=KITgreen,
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draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.7, 0.7, 1) {};
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\node[color=KITgreen, right=0cm of f] {$\boldsymbol{f}$};
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\end{tikzpicture}
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\caption{$\text{poly}\left( \mathcal{C} \right)$ for $n=3$}
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\end{figure}
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\end{minipage}
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\end{frame}
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\begin{frame}[t]
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\frametitle{ADMM Decoding: Algorithm}
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\todo{TODO}
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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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%
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% \begin{minipage}[c]{0.6\linewidth}
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% \begin{itemize}
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% \item Set of all variable nodes incident to a check node:
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% \begin{align*}
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% N\left( j \right) \equiv \left\{
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% i | i\in \mathcal{I},
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% \boldsymbol{H}_{j,i} = 1
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% \right\},
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% j \in \mathcal{J}
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% \end{align*}
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% \begin{align*}
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% S \subseteq N\left( j \right), \left| S \right| \text{odd}
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% \end{align*}
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% \item Relaxed polytope representation:
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% \begin{align*}
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% \sum_{i\in \left( N\left( j \right) \setminus S\right) } f_i
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% + \sum_{i\in S} \left( 1 - f_i \right) \ge 1
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% \end{align*}
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% ``$\boldsymbol{f}$ is separated by at least one bitflip
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% from all illegal configurations''
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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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%
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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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%
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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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%
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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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%
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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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%
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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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%
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% % Codeword Polytope
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%
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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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%
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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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%
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% \draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);
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%
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% % Polytope Annotations
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%
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% \node[codeword, color=KITred] (c111) at (2, 2, 2) {};% {$\left( 0, 0, 0 \right) $};
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% \node[codeword, color=KITred] (c001) at (0, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
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% \node[codeword, color=KITred] (c100) at (2, 0, 0) {};% {$\left( 1, 1, 0 \right) $};
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% \node[codeword, color=KITred] (c010) at (0, 2, 0) {};% {$\left( 0, 1, 1 \right) $};
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%
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% \node[color=KITred, left=0cm of c111] {$\left( 1, 1, 1 \right) $};
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% \node[color=KITred, right=0cm of c001] {$\left( 0, 0, 1 \right) $};
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% \node[color=KITred, right=0.35cm of c100] {$\left( 1, 0, 0 \right) $};
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% \node[color=KITred, below=0cm of c010] {$\left( 0, 1, 0 \right) $};
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% \end{tikzpicture}
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% \caption{Relaxed polytope for $n=3$}
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% \end{figure}
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% \end{minipage}
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% \todo{How is this a relaxation and not just an alternative formulation?
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% We have just switched out valid codewords for invalid ones}
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% \todo{Is LP Relaxation relevant as theoretical background?}
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%\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{ADMM}%
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%\label{sub:Alg ADMM}
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%
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%\begin{frame}[t]
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% \frametitle{ADMM Decoding: General Idea}
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%
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% \todo{TODO}
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%\end{frame}
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%
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\begin{frame}[t]
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% \frametitle{ADMM Decoding: Algorithm}
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%
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% \todo{TODO}
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%\end{frame}
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@ -1438,13 +1438,13 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{x}}_n\text{ with lowest }d
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%\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{ADMM: Examination Results}%
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\label{sub:Ex ADMM}
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\begin{frame}[t]
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\frametitle{ADMM}
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\todo{TODO}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{ADMM: Examination Results}%
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%\label{sub:Ex ADMM}
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%
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%\begin{frame}[t]
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% \frametitle{ADMM}
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%
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% \todo{TODO}
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%\end{frame}
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@ -3,12 +3,28 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{TODO}%
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\label{sub:TODO}
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\subsection{LP Decoding}%
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\label{sub:LP Decoding}
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\begin{frame}[t]
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\frametitle{TODO}
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\frametitle{Forthcoming Examination: LP Decoding}
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\begin{itemize}
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\item Test the (Alternating Direction Method of Multipliers) ADMM
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as an optimization method for LP Decoding
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\begin{itemize}
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\item ADMM is intended to blend the decomposability
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of dual ascent with the superior convergence properties of the method
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of multipliers \cite{distr_opt_book}
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\item Recently, ADMM has been proposed for efficient LP Decoding
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\cite{efficient_lp_dec_admm}
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\end{itemize}
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\item Compare ADMM implementation with Proximal Decoding implementation with respect to
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\begin{itemize}
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\item Decoding performance (BER, FER)
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\item Computational performance (time complexity, actual seconds per frame)
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\end{itemize}
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\end{itemize}
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\todo{TODO}
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\end{frame}
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@ -105,35 +105,20 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{LP Decoding}%
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\label{sub:LP Decoding}
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\subsection{Optimization as a Decoding Method}%
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\label{sub:Optimization as a Decoding Method}
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\begin{frame}[t]
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\frametitle{LP Decoding}
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\frametitle{Optimization as a Decoding Method}
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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}}\lambda_{\boldsymbol{c}}
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\boldsymbol{c} : \lambda_{\boldsymbol{c}} \ge 0,
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\sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}} = 1
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\right\},
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\hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R}
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\end{align*}
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\item Cost Function:
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\begin{align*}
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\sum_{i=1}^{n} \gamma_i c_i,
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\hspace{5mm}\gamma_i = \log\left(
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\frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
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\end{align*}
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\item LP Formulation of ML Decoding:
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\begin{align*}
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&\text{minimize } \sum_{i=1}^{n} \gamma_i f_i \\
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&\text{subject to } \boldsymbol{f}\in\text{poly}\left( \mathcal{C} \right)
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\end{align*}
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\item Reormulate decoding problem as optimization problem
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\begin{itemize}
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\item Establish objective function
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\item Establish constraints
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\end{itemize}
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\item Use optimization method to solve the new problem
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\end{itemize}
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\end{minipage}%
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\hfill%
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@ -165,14 +150,14 @@
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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] (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] (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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% \draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);
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% Polytope Annotations
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@ -192,92 +177,9 @@
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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] {$\boldsymbol{f}$};
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\end{tikzpicture}
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\caption{$\text{poly}\left( \mathcal{C} \right)$ for $n=3$}
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\caption{Hypercube ($n=3$) with valid codewords}
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\end{figure}
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\end{minipage}
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\todo{Move this slide to LP decoding}
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\end{frame}
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\begin{frame}[t]
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\frametitle{LP Relaxation}
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\begin{minipage}[c]{0.6\linewidth}
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\begin{itemize}
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\item Set of all variable nodes incident to a check node:
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\begin{align*}
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N\left( j \right) \equiv \left\{
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i | i\in \mathcal{I},
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\boldsymbol{H}_{j,i} = 1
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\right\},
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j \in \mathcal{J}
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\end{align*}
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\begin{align*}
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S \subseteq N\left( j \right), \left| S \right| \text{odd}
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\end{align*}
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\item Relaxed polytope representation:
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\begin{align*}
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\sum_{i\in \left( N\left( j \right) \setminus S\right) } f_i
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+ \sum_{i\in S} \left( 1 - f_i \right) \ge 1
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\end{align*}
|
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``$\boldsymbol{f}$ is separated by at least one bitflip
|
||||
from all illegal configurations''
|
||||
\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, color=KITred] (c111) at (2, 2, 2) {};% {$\left( 0, 0, 0 \right) $};
|
||||
\node[codeword, color=KITred] (c001) at (0, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
|
||||
\node[codeword, color=KITred] (c100) at (2, 0, 0) {};% {$\left( 1, 1, 0 \right) $};
|
||||
\node[codeword, color=KITred] (c010) at (0, 2, 0) {};% {$\left( 0, 1, 1 \right) $};
|
||||
|
||||
\node[color=KITred, left=0cm of c111] {$\left( 1, 1, 1 \right) $};
|
||||
\node[color=KITred, right=0cm of c001] {$\left( 0, 0, 1 \right) $};
|
||||
\node[color=KITred, right=0.35cm of c100] {$\left( 1, 0, 0 \right) $};
|
||||
\node[color=KITred, below=0cm of c010] {$\left( 0, 1, 0 \right) $};
|
||||
\end{tikzpicture}
|
||||
\caption{Relaxed polytope for $n=3$}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\todo{How is this a relaxation and not just an alternative formulation?
|
||||
We have just switched out valid codewords for invalid ones}
|
||||
\todo{Is LP Relaxation relevant as theoretical background?}
|
||||
\end{frame}
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user