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Added content to LP Relaxation slide

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Andreas Tsouchlos 2022-12-20 14:56:59 +01:00
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latex/presentations/midterm/sections/theoretical_background.tex
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@ -28,7 +28,8 @@
\begin{tikzpicture}[scale=1, transform shape]
\node (in) {$c\left[ k \right] $};
\node[mapper, right=0.5cm of in] (bpskmap) {Mapper};
\node[right=1.5cm of bpskmap, draw, circle, inner sep=0pt, minimum size=0.5cm] (add) {$+$};
\node[right=1.5cm of bpskmap,
draw, circle, inner sep=0pt, minimum size=0.5cm] (add) {$+$};
\node[right=0.5cm of add] (out) {$y\left[ k \right] $};
\node[below=0.5cm of add] (noise) {$n\left[ k \right] $};
@ -76,18 +77,19 @@
\begin{align*}
\text{poly}\left( \mathcal{C} \right) =
\left\{
\sum_{\boldsymbol{c}\in\mathcal{C}} \lambda_{\boldsymbol{c}} \boldsymbol{c}
: \lambda_{\boldsymbol{c}} \ge 0,
\sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}}
\boldsymbol{c} : \lambda_{\boldsymbol{c}} \ge 0,
\sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}} = 1
\right\},
\hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R}
\end{align*}
\item Cost Function:
\begin{align*}
\gamma_i = \log\left( \frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right),
\hspace{5mm} i = \left\{ 1, \ldots, n \right\}
\sum_{i=1}^{n} \gamma_i c_i,
\hspace{5mm}\gamma_i = \log\left(
\frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
\end{align*}
\item LP Formulation:
\item LP Formulation of ML Decoding:
\begin{align*}
&\text{minimize } \sum_{i=1}^{n} \gamma_i f_i \\
&\text{subject to } \boldsymbol{f}\in\text{poly}\left( \mathcal{C} \right)
@ -157,7 +159,81 @@
\begin{frame}[t]
\frametitle{LP Relaxation}
\todo{TODO}
\begin{minipage}[c]{0.6\linewidth}
\begin{itemize}
\item Set of all variable nodes incident to a check node:
\begin{align*}
N\left( j \right) \equiv \left\{
i | i\in \mathcal{I},
\boldsymbol{H}_{i,j} = 1
\right\},
j \in \mathcal{J}
\end{align*}
\item ``Illegal configurations''
\begin{align*}
S \subseteq N\left( j \right), \left| S \right| \text{odd}
\end{align*}
\item Relaxed polytope representation:
\begin{align*}
\sum_{i\in \left( N\left( j \right) \setminus S\right) }f_i + \sum_{i\in S} \left( 1 - f_i \right) \ge 1
\end{align*}
``$\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?}
\end{frame}