722 lines
30 KiB
TeX
722 lines
30 KiB
TeX
\section{Comparison}%
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\label{sec:Comparison}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Comparison of Simulation Results}%
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\label{sub:Comparison of Simulation Results}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Comparison: Convergence Behavior}
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\vspace*{-3mm}
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\begin{itemize}
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\item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
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\end{itemize}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={Iterations},
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ylabel={Average $\lVert \boldsymbol{c}-\boldsymbol{\hat{c}} \rVert$},
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width=0.9\textwidth,
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height=0.675\textwidth,
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%legend pos=outer north east,
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]
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\addplot [ForestGreen, mark=none, line width=1pt]
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table [col sep=comma,
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discard if not={omega}{0.0774263682681127},
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discard if gt={k}{101},
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x=k, y=err]
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{res/proximal/2d_avg_error_20433484_1db.csv};
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\addlegendentry{$E_b / N_0 = \SI{1}{dB}$}
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\addplot [NavyBlue, mark=none, line width=1pt]
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table [col sep=comma, discard if not={omega}{0.0774263682681127},
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discard if gt={k}{101},
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x=k, y=err]
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{res/proximal/2d_avg_error_20433484_3db.csv};
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\addlegendentry{$E_b / N_0 = \SI{3}{dB}$}
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\addplot [RedOrange, mark=none, line width=1pt]
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table [col sep=comma, discard if not={omega}{0.052233450742668434},
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discard if gt={k}{101},
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x=k, y=err]
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{res/proximal/2d_avg_error_20433484_5db.csv};
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\addlegendentry{$E_b / N_0 = \SI{5}{dB}$}
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\addplot [RoyalPurple, mark=none, line width=1pt]
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table [col sep=comma, discard if not={omega}{0.052233450742668434},
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discard if gt={k}{101},
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x=k, y=err]
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{res/proximal/2d_avg_error_20433484_8db.csv};
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\addlegendentry{$E_b / N_0 = \SI{8}{dB}$}
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\end{axis}
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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width=0.9\textwidth,
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height=0.675\textwidth,
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% legend pos=outer north east,
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xlabel={Iteration},
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ylabel={Average $\left\Vert \hat{\boldsymbol{c}} - \boldsymbol{c} \right\Vert$}
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]
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\addplot[ForestGreen, line width=1pt]
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table [col sep=comma, x=k, y=err,
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discard if not={SNR}{1.0},
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discard if gt={k}{101}]
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{res/admm/avg_error_20433484.csv};
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\addlegendentry{$E_b / N_0 = \SI{1}{dB}$}
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\addplot[RedOrange, line width=1pt]
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table [col sep=comma, x=k, y=err,
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discard if not={SNR}{2.0},
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discard if gt={k}{101}]
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{res/admm/avg_error_20433484.csv};
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\addlegendentry{$E_b / N_0 = \SI{2}{dB}$}
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\addplot[NavyBlue, line width=1pt]
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table [col sep=comma, x=k, y=err,
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discard if not={SNR}{3.0},
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discard if gt={k}{101}]
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{res/admm/avg_error_20433484.csv};
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\addlegendentry{$E_b / N_0 = \SI{3}{dB}$}
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\addplot[RoyalPurple, line width=1pt]
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table [col sep=comma, x=k, y=err,
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discard if not={SNR}{4.0},
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discard if gt={k}{101}]
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{res/admm/avg_error_20433484.csv};
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\addlegendentry{$E_b / N_0 = \SI{4}{dB}$}
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\end{axis}
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\end{tikzpicture}
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\end{subfigure}%
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\end{figure}
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\begin{itemize}
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\item Minimum number of iterations independant of SNR
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\end{itemize}
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\bigskip
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\smallskip
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\smallskip
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\smallskip
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\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
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Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Comparison: Time Complexity}
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\vspace*{-6.5mm}
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\begin{itemize}
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\item Codes: BCH $\left( 31, 11 \right)$; BCH $\left( 31, 26 \right)$;
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\citereference{Mac23, 96.3.965; 204.33.484; 204.55.187; 408.33.844; PEGReg252x504}
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\item Measured performance: $\sim\SI{10000}{}$ frames/s
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on Intel Core i7-7700HQ @ 2.80GHz; $n=204$
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\item Both algorithms are $\mathcal{O}\left( n \right)$ on average
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\end{itemize}
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$n$}, ylabel={time per frame (s)},
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legend style={at={(0.05,0.6)},anchor=south west},
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legend cell align={left},
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width=0.43\textwidth,
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height=0.32\textwidth,
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]
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\addplot[RedOrange, only marks, mark=square*]
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table [col sep=comma, x=n, y=spf]
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{res/proximal/fps_vs_n.csv};
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\addlegendentry{proximal}
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\addplot[Gray, only marks, mark=*]
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table [col sep=comma, x=n, y=spf]
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{res/hybrid/fps_vs_n.csv};
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\addlegendentry{improved ($\SI{12}{\bit}$)}
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\addplot[NavyBlue, only marks, mark=triangle*]
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table [col sep=comma, x=n, y=spf]
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{res/admm/fps_vs_n.csv};
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\addlegendentry{ADMM}
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\end{axis}
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\end{tikzpicture}
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\end{figure}
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\bigskip
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\smallskip
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\smallskip
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\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
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Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Comparison: Decoding Performance}
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\vspace{-0.5cm}
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\begin{itemize}
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\item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
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\end{itemize}
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={BER},
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ymode=log,
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legend columns=2,
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legend style={at={(0.5,-0.45)},anchor=south},
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ymax=1.5, ymin=3e-8,
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width=0.45\textwidth,
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height=0.33\textwidth,
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]
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\addplot[RedOrange, mark=*, line width=1pt]
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table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot[RedOrange, mark=triangle, densely dashed, line width=1pt]
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table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
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{res/hybrid/2d_ber_fer_dfr_20433484.csv};
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\addplot[NavyBlue, line width=1pt, mark=*]
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table [x=SNR, y=BER, col sep=comma,
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discard if not={mu}{3.0},
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%discard if gt={SNR}{4.0}
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]
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{res/admm/ber_2d_20433484.csv};
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\addplot [RoyalPurple, mark=*, line width=1pt]
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table [x=SNR, y=BP, col sep=comma] {res/ber_paper.csv};
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\end{axis}
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\end{tikzpicture}
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={FER},
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ymode=log,
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legend columns=2,
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legend style={at={(0.5,-0.45)},anchor=south},
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ymax=1.5, ymin=3e-8,
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width=0.45\textwidth,
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height=0.33\textwidth,
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]
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\addplot[RedOrange, mark=*, solid, line width=1pt]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot[RedOrange, mark=triangle, densely dashed, line width=1pt]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/hybrid/2d_ber_fer_dfr_20433484.csv};
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\addplot[NavyBlue, line width=1pt, mark=*]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={mu}{3.0},
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%discard if gt={SNR}{4.0}
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]
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{res/admm/ber_2d_20433484.csv};
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\addplot[Black, line width=1pt, mark=*]
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table [col sep=comma, x=SNR, y=FER,
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%discard if gt={SNR}{3.0}
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]
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{res/generic/fer_ml_20433484.csv};
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\end{axis}
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\end{tikzpicture}
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\begin{tikzpicture}
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\begin{axis}[hide axis,
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xmin=10, xmax=50,
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ymin=0, ymax=0.4,
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legend columns=3,
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legend style={draw=white!15!black,legend cell align=left}]
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\addlegendimage{RedOrange, mark=*, line width=1pt}
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\addlegendentry{Proximal}
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\addlegendimage{NavyBlue, mark=*, line width=1pt}
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\addlegendentry{ADMM}
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\addlegendimage{Black, mark=*, line width=1pt}
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\addlegendentry{ML}
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\addlegendimage{RedOrange, mark=triangle, densely dashed, line width=1pt}
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\addlegendentry{Improved ($N$=12)}
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\addlegendimage{RoyalPurple, mark=*, line width=1pt}
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\addlegendentry{BP}
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\end{axis}
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\end{tikzpicture}
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\end{figure}
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\smallskip
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\smallskip
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\smallskip
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\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
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Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\begin{frame}[t]
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% \frametitle{Decoding Performance}
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%
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% \vspace*{-5mm}
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%
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% \begin{figure}[H]
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% \centering
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%
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% \begin{tikzpicture}
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% \begin{axis}[
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% grid=both,
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% xlabel={$E_b / N_0 \left( \text{dB} \right) $}, ylabel={FER},
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% ymode=log,
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% ymax=1.5, ymin=5e-5,
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% legend columns = 3,
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% legend style={at={(0.5,-0.5)},anchor=south},
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% width=0.45\textwidth,
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% height=0.3375\textwidth,
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% ]
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%
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% \addplot[RedOrange, line width=1pt, mark=*, solid]
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% table [x=SNR, y=FER, col sep=comma,
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% discard if not={gamma}{0.05},
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% discard if gt={SNR}{5.5}]
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% {res/proximal/2d_ber_fer_dfr_20433484.csv};
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% \addlegendentry{Proximal decoding}
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% \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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% table [x=SNR, y=FER, col sep=comma,
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% discard if not={mu}{3.0},
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% discard if gt={SNR}{4.0}]
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% {res/admm/ber_2d_20433484.csv};
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% \addlegendentry{ADMM}
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% \addplot[PineGreen, line width=1pt, mark=triangle, solid]
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% table [col sep=comma, x=SNR, y=FER,
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% discard if gt={SNR}{3.0}]
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% {res/generic/fer_ml_20433484.csv};
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% \addlegendentry{ML}
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% \end{axis}
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% \end{tikzpicture}
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%
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% \caption{Comparison of FER of proximal decoding and LP decoding using ADMM%
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% \protect\footnotemark{}}
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% \end{figure}%
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%
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% \footnotetext{Simulation performed with (3,6) regular LDPC code with $n=204, k=102$
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% \cite[Code: 204.33.484]{mackay_enc}}
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%\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Comparison: Decoding Performance}
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\captionsetup[subfigure]{font=footnotesize}
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\hspace*{-0.4cm}
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\begin{minipage}[c]{0.9\textwidth}
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\centering
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\begin{figure}[H]
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\vspace*{-0.8cm}
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\centering
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\begin{subfigure}[t]{0.33\textwidth}
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\centering
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\begin{tikzpicture}[scale=0.8]
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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width=1.2\textwidth,
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height=0.825\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_963965.csv};
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\addplot[RedOrange, mark=triangle, densely dashed, line width=1pt]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.05}, discard if gt={SNR}{5.5}]
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{res/hybrid/2d_ber_fer_dfr_963965.csv};
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\addplot[NavyBlue, line width=1pt, mark=*]
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table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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%{res/hybrid/2d_ber_fer_dfr_963965.csv};
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{res/admm/ber_2d_963965.csv};
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\addplot[Black, line width=1pt, mark=*]
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table [col sep=comma, x=SNR, y=FER,]
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{res/generic/fer_ml_9633965.csv};
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\end{axis}
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\end{tikzpicture}
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\vspace*{-2mm}
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\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$
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\citereference{Mac23, 96.33.965}}
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\end{subfigure}%
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\begin{subfigure}[t]{0.33\textwidth}
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\centering
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\begin{tikzpicture}[scale=0.8]
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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width=1.2\textwidth,
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height=0.825\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.05},
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discard if gt={SNR}{5.5}]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot[RedOrange, mark=triangle, densely dashed, line width=1pt]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.05}, discard if gt={SNR}{5.5}]
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{res/hybrid/2d_ber_fer_dfr_20433484.csv};
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\addplot[NavyBlue, line width=1pt, mark=*]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={mu}{3.0},
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discard if gt={SNR}{5.5}]
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{res/admm/ber_2d_20433484.csv};
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\addplot[Black, line width=1pt, mark=*]
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table [col sep=comma, x=SNR, y=FER,
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discard if gt={SNR}{5.5}]
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{res/generic/fer_ml_20433484.csv};
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\end{axis}
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\end{tikzpicture}
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\vspace*{-2mm}
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\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$
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\citereference{Mac23, 204.33.484}}
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\end{subfigure}%
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\begin{subfigure}[t]{0.33\textwidth}
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\centering
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\begin{tikzpicture}[scale=0.8]
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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width=1.2\textwidth,
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height=0.825\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_40833844.csv};
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\addplot[RedOrange, mark=triangle, densely dashed, line width=1pt]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.05}, discard if gt={SNR}{5.5}]
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{res/hybrid/2d_ber_fer_dfr_40833844.csv};
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\addplot[NavyBlue, line width=1pt, mark=*]
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table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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{res/admm/ber_2d_40833844.csv};
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\end{axis}
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\end{tikzpicture}
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\vspace*{-2mm}
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\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$
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\citereference{Mac23, 408.33.844}}
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\end{subfigure}%
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|
|
\vspace*{-2mm}
|
|
|
|
\begin{subfigure}[t]{0.33\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}[scale=0.8]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
|
|
ymode=log,
|
|
ymax=1.5, ymin=8e-5,
|
|
width=1.2\textwidth,
|
|
height=0.825\textwidth,
|
|
]
|
|
|
|
\addplot[RedOrange, line width=1pt, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
|
|
\addplot[RedOrange, mark=triangle, densely dashed, line width=1pt]
|
|
table [x=SNR, y=FER, col sep=comma,
|
|
discard if not={gamma}{0.05}, discard if gt={SNR}{5.5}]
|
|
{res/hybrid/2d_ber_fer_dfr_bch_31_26.csv};
|
|
\addplot[NavyBlue, line width=1pt, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
|
|
{res/admm/ber_2d_bch_31_26.csv};
|
|
\addplot[Black, line width=1pt, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma,
|
|
discard if gt={SNR}{5.5},
|
|
discard if lt={SNR}{1},
|
|
]
|
|
{res/generic/fer_ml_bch_31_26.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\vspace*{-2mm}
|
|
|
|
\caption{BCH code with $n=31, k=26$}
|
|
\end{subfigure}%
|
|
\begin{subfigure}[t]{0.33\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}[scale=0.8]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
|
|
ymode=log,
|
|
ymax=1.5, ymin=8e-5,
|
|
width=1.2\textwidth,
|
|
height=0.825\textwidth,
|
|
]
|
|
|
|
\addplot[RedOrange, line width=1pt, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_20455187.csv};
|
|
\addplot[RedOrange, mark=triangle, densely dashed, line width=1pt]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/hybrid/2d_ber_fer_dfr_20455187.csv};
|
|
\addplot[NavyBlue, line width=1pt, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
|
|
{res/admm/ber_2d_20455187.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\vspace*{-2mm}
|
|
|
|
\caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$
|
|
\citereference{Mac23, 204.55.187}}
|
|
\end{subfigure}%
|
|
\begin{subfigure}[t]{0.33\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}[scale=0.8]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
|
|
ymode=log,
|
|
ymax=1.5, ymin=8e-5,
|
|
width=1.2\textwidth,
|
|
height=0.825\textwidth,
|
|
]
|
|
|
|
\addplot[RedOrange, line width=1pt, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\addplot[RedOrange, mark=triangle, densely dashed, line width=1pt]
|
|
table [x=SNR, y=FER, col sep=comma,
|
|
discard if not={gamma}{0.05}, discard if gt={SNR}{5.5}]
|
|
{res/hybrid/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\addplot[NavyBlue, line width=1pt, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
|
|
{res/admm/ber_2d_pegreg252x504.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\vspace*{-2mm}
|
|
|
|
\caption{LDPC code (progressive edge growth construction) with $n=504, k=252$
|
|
\citereference{Mac23, PEGReg252x504}}
|
|
\end{subfigure}%
|
|
\end{figure}
|
|
\end{minipage}%
|
|
\begin{minipage}[c]{0.1\textwidth}
|
|
\centering
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\vspace*{-1.2cm}
|
|
\hspace*{2mm}
|
|
\begin{tikzpicture}[scale=0.8]
|
|
\begin{axis}[hide axis,
|
|
xmin=10, xmax=50,
|
|
ymin=0, ymax=0.4,
|
|
legend columns=1,
|
|
legend style={draw=white!15!black}]
|
|
\addlegendimage{RedOrange, line width=1pt, mark=*, solid}
|
|
\addlegendentry{Proximal}
|
|
|
|
\addlegendimage{RedOrange, line width=1pt, mark=triangle*, densely dashed}
|
|
\addlegendentry{Improved}
|
|
|
|
\addlegendimage{NavyBlue, line width=1pt, mark=*}
|
|
\addlegendentry{ADMM}
|
|
|
|
\addlegendimage{Black, line width=1pt, mark=*}
|
|
\addlegendentry{ML}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{figure}
|
|
\end{minipage}
|
|
|
|
\vspace*{-1mm}
|
|
|
|
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
|
|
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
|
|
\end{frame}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Theoretical Comparison}%
|
|
\label{sub:Theoretical Comparison}
|
|
|
|
\begin{frame}[t, fragile]
|
|
\frametitle{Comparison: Algorithm Structure}
|
|
|
|
\vspace*{-1cm}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
|
|
\begin{subfigure}[b]{0.47\textwidth}
|
|
\centering
|
|
|
|
\begin{align*}
|
|
\text{minimize}\hspace{2mm} & \textcolor{KITblue}{\underbrace{L\left(
|
|
\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)}_{\text{Likelihood}}}
|
|
+ \textcolor{KITred}{\underbrace{\gamma h\left( \tilde{\boldsymbol{x}}
|
|
\right)} _{\text{Constraints}}} \\
|
|
\text{subject to}\hspace{2mm} &\tilde{\boldsymbol{x}} \in \mathbb{R}^n
|
|
\end{align*}
|
|
|
|
\vspace*{-5mm}
|
|
|
|
\begin{algorithm}[caption={}, label={},
|
|
basicstyle=\fontsize{10}{18}\selectfont
|
|
]
|
|
Initialize $\boldsymbol{r}, \boldsymbol{s}, \omega, \gamma$
|
|
while stopping critierion unfulfilled do
|
|
$\textcolor{KITblue}{\boldsymbol{r} \leftarrow \boldsymbol{r}
|
|
+ \omega \nabla L\left( \boldsymbol{y} \mid \boldsymbol{s} \right)} $
|
|
$\textcolor{KITred}{\boldsymbol{s} \leftarrow
|
|
\textbf{prox}_{\gamma h}\left( \boldsymbol{r} \right)} $|\Suppressnumber|
|
|
|\Reactivatenumber|
|
|
end while
|
|
return $\boldsymbol{s}$
|
|
\end{algorithm}
|
|
\end{subfigure}\hfill%
|
|
\begin{subfigure}[b]{0.5\textwidth}
|
|
\centering
|
|
|
|
\begin{align*}
|
|
\text{minimize}\hspace{5mm} &
|
|
\textcolor{KITblue}{\underbrace{\boldsymbol{\gamma}^\text{T}
|
|
\tilde{\boldsymbol{c}}}_{\text{Likelihood}}}
|
|
+ \textcolor{KITred}{\underbrace{\sum\nolimits_{j\in\mathcal{J}}
|
|
g_j\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
|
|
\right)}_{\text{Constraints}}} \\
|
|
\text{subject to}\hspace{5mm} &
|
|
\tilde{\boldsymbol{c}} \in \left[ 0, 1 \right]^n
|
|
\end{align*}
|
|
|
|
\vspace*{-5mm}
|
|
|
|
\begin{algorithm}[caption={}, label={},
|
|
basicstyle=\fontsize{10}{18}\selectfont
|
|
]
|
|
Initialize $\tilde{\boldsymbol{c}}, \boldsymbol{z}, \boldsymbol{u}, \boldsymbol{\gamma}, \rho$
|
|
while stopping criterion unfulfilled do
|
|
$\textcolor{KITblue}{\tilde{\boldsymbol{c}} \leftarrow \argmin_{\tilde{\boldsymbol{c}}}
|
|
\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
|
|
+ \frac{\rho}{2}\sum_{j\in\mathcal{J}} \left\Vert
|
|
\boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j
|
|
+ \boldsymbol{u}_j \right\Vert \right)}$
|
|
$\textcolor{KITred}{\boldsymbol{z}_j \leftarrow \textbf{prox}_{g_j}
|
|
\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
|
|
+ \boldsymbol{u}_j \right), \hspace{5mm}\forall j\in\mathcal{J}}$
|
|
$\boldsymbol{u}_j \leftarrow \boldsymbol{u}_j
|
|
+ \tilde{\boldsymbol{c}} - \boldsymbol{z}_j, \hspace{14.5mm}\forall j\in\mathcal{J}$
|
|
end while
|
|
return $\tilde{\boldsymbol{c}}$
|
|
\end{algorithm}
|
|
\end{subfigure}%
|
|
\end{figure}%
|
|
|
|
\end{frame}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t, fragile]
|
|
\frametitle{Comparison: Algorithm Structure}
|
|
|
|
\vspace*{-1cm}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
|
|
\begin{subfigure}{0.48\textwidth}
|
|
\centering
|
|
|
|
\begin{algorithm}[caption={}, label={},]
|
|
Initialize $\boldsymbol{r}, \boldsymbol{s}, \omega, \gamma$
|
|
while stopping critierion unfulfilled do
|
|
for j in $\mathcal{J}$ do
|
|
$p_j \leftarrow \prod_{i\in N_c\left( j \right) } r_i $
|
|
$\textcolor{KITblue}{M_{j\to i} \leftarrow p_j^2 - p_j}$|\Suppressnumber|
|
|
|\vspace{0.7mm}\Reactivatenumber|
|
|
end for
|
|
for i in $\mathcal{I}$ do
|
|
$s_i \leftarrow s_i + \gamma \left[ 4\left( s_i^2 - 1 \right)s_i
|
|
\phantom{\frac{4}{s_i}}\right.$|\Suppressnumber|
|
|
|\Reactivatenumber|$\left.+ \frac{4}{s_i}\sum_{j\in N_v\left( i \right) }
|
|
M_{j\to i} \right] $
|
|
$r_i \leftarrow r_i + \omega \left( s_i - y_i \right)$
|
|
end for
|
|
end while
|
|
return $\boldsymbol{s}$
|
|
\end{algorithm}
|
|
\end{subfigure}%
|
|
\hfill
|
|
\begin{subfigure}{0.48\textwidth}
|
|
\centering
|
|
|
|
\begin{algorithm}[caption={}, label={},]
|
|
Initialize $\tilde{\boldsymbol{c}}, \boldsymbol{z}, \boldsymbol{u}, \boldsymbol{\gamma}, \rho$
|
|
while stopping criterion unfulfilled do
|
|
for j in $\mathcal{J}$ do
|
|
$\boldsymbol{z}_j \leftarrow \Pi_{P_{d_j}}\left(
|
|
\boldsymbol{T}_j\tilde{\boldsymbol{c}} + \boldsymbol{u}_j\right)$
|
|
$\boldsymbol{u}_j \leftarrow \boldsymbol{u}_j + \boldsymbol{T}_j\tilde{\boldsymbol{c}}
|
|
- \boldsymbol{z}_j$
|
|
$\textcolor{KITblue}{M_{j\to i} \leftarrow \left( z_j \right)_i - \left( u_j \right)_i,
|
|
\hspace{3mm} \forall i \in N_c\left( j \right)}$
|
|
end for
|
|
for i in $\mathcal{I}$ do
|
|
$\tilde{c}_i \leftarrow \frac{1}{d_i}
|
|
\left(\sum_{j\in N_v\left( i \right) } M_{j\to i}
|
|
- \frac{\gamma_i}{\mu} \right)$|\Suppressnumber|
|
|
|\vspace{7mm}\Reactivatenumber|
|
|
end for
|
|
end while
|
|
return $\tilde{\boldsymbol{c}}$
|
|
\end{algorithm}
|
|
\end{subfigure}%
|
|
\end{figure}%
|
|
\end{frame}
|
|
|