From a9bbbd4978f115265e3c1a6774d8ec34ab120ebd Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Wed, 19 Apr 2023 22:53:45 +0200 Subject: [PATCH] Moved all citations to the respective slides --- latex/presentations/final/presentation.tex | 9 +- .../presentations/final/sections/appendix.tex | 82 +++++---- .../final/sections/comparison.tex | 120 +++++++------ .../final/sections/conclusion.tex | 26 +++ .../final/sections/lp_dec_using_admm.tex | 167 ++++++++++-------- .../final/sections/proximal_decoding.tex | 106 ++++++++--- .../final/sections/theoretical_background.tex | 38 +++- 7 files changed, 344 insertions(+), 204 deletions(-) create mode 100644 latex/presentations/final/sections/conclusion.tex diff --git a/latex/presentations/final/presentation.tex b/latex/presentations/final/presentation.tex index 435a444..04174c7 100644 --- a/latex/presentations/final/presentation.tex +++ b/latex/presentations/final/presentation.tex @@ -300,12 +300,13 @@ \input{sections/proximal_decoding.tex} \input{sections/lp_dec_using_admm.tex} \input{sections/comparison.tex} + \input{sections/conclusion.tex} \input{sections/question_slide.tex} - \begin{frame}[allowframebreaks] - \frametitle{Bibliography} - \printbibliography[heading=none] - \end{frame} +% \begin{frame}[allowframebreaks] +% \frametitle{Bibliography} +% \printbibliography[heading=none] +% \end{frame} \input{sections/appendix.tex} diff --git a/latex/presentations/final/sections/appendix.tex b/latex/presentations/final/sections/appendix.tex index 8abf463..155ae2f 100644 --- a/latex/presentations/final/sections/appendix.tex +++ b/latex/presentations/final/sections/appendix.tex @@ -27,7 +27,7 @@ ylabel={$\gamma$}, zlabel={BER}, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, legend pos=outer north east,] \addplot3[surf, @@ -50,10 +50,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$ - \cite[\text{96.3.965}]{mackay_enc}} + \citereference{Mac23, 96.3.965}} \end{subfigure}% \begin{subfigure}[t]{0.33\textwidth} \centering @@ -65,7 +65,7 @@ ylabel={$\gamma$}, zlabel={BER}, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, legend pos=outer north east,] \addplot3[surf, @@ -88,10 +88,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$ - \cite[\text{204.33.484}]{mackay_enc}} + \citereference{Mac23, 204.33.484}} \end{subfigure}% \begin{subfigure}[t]{0.33\textwidth} \centering @@ -103,7 +103,7 @@ ylabel={$\gamma$}, zlabel={BER}, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, legend pos=outer north east,] \addplot3[surf, @@ -126,12 +126,14 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$ - \cite[\text{408.33.844}]{mackay_enc}} + \citereference{Mac23, 408.33.844}} \end{subfigure}% + \vspace*{-2mm} + \begin{subfigure}[t]{0.33\textwidth} \centering @@ -142,7 +144,7 @@ ylabel={$\gamma$}, zlabel={BER}, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, legend pos=outer north east,] \addplot3[surf, @@ -165,7 +167,7 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{BCH code with $n=31, k=26$} \end{subfigure}% @@ -179,7 +181,7 @@ ylabel={$\gamma$}, zlabel={BER}, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, legend pos=outer north east,] \addplot3[surf, @@ -202,10 +204,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$ - \cite[\text{204.55.187}]{mackay_enc}} + \citereference{Mac23, 204.55.187}} \end{subfigure}% \begin{subfigure}[t]{0.33\textwidth} \centering @@ -217,7 +219,7 @@ ylabel={$\gamma$}, zlabel={BER}, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, legend pos=outer north east,] \addplot3[surf, @@ -240,10 +242,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$ - \cite[\text{PEGReg252x504}]{mackay_enc}} + \citereference{Mac23, PEGReg252x504}} \end{subfigure}% \end{figure} \end{minipage}% @@ -271,6 +273,11 @@ \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -296,7 +303,7 @@ xlabel={$\rho$}, ylabel={Avg. \# iter.}, ymode=log, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[ForestGreen, line width=1pt] table [col sep=comma, x=rho, y=k_avg, @@ -313,10 +320,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$ - \cite[\text{96.3.965}]{mackay_enc}} + \citereference{Mac23, 96.3.965}} \end{subfigure}% \begin{subfigure}[t]{0.33\textwidth} \centering @@ -327,7 +334,7 @@ xlabel={$\rho$}, ylabel={Avg. \# iter.}, ymode=log, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[ForestGreen, line width=1pt] table [col sep=comma, x=rho, y=k_avg, @@ -344,10 +351,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$ - \cite[\text{204.33.484}]{mackay_enc}} + \citereference{Mac23, 204.33.484}} \end{subfigure}% \begin{subfigure}[t]{0.33\textwidth} \centering @@ -358,7 +365,7 @@ xlabel={$\rho$}, ylabel={Avg. \# iter.}, ymode=log, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[ForestGreen, line width=1pt] table [col sep=comma, x=rho, y=k_avg, @@ -375,12 +382,14 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$ - \cite[\text{408.33.844}]{mackay_enc}} + \citereference{Mac23, 408.33.844}} \end{subfigure}% + \vspace*{-2mm} + \begin{subfigure}[t]{0.33\textwidth} \centering @@ -390,7 +399,7 @@ xlabel={$\rho$}, ylabel={Avg. \# iter.}, ymode=log, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[ForestGreen, line width=1pt] table [col sep=comma, x=rho, y=k_avg, @@ -407,7 +416,7 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{BCH code with $n=31, k=26$} \end{subfigure}% @@ -420,7 +429,7 @@ xlabel={$\rho$}, ylabel={Avg. \# iter.}, ymode=log, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[ForestGreen, line width=1pt] table [col sep=comma, x=rho, y=k_avg, @@ -437,10 +446,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$ - \cite[\text{204.55.187}]{mackay_enc}} + \citereference{Mac23, 204.55.187}} \end{subfigure}% \begin{subfigure}[t]{0.33\textwidth} \centering @@ -451,7 +460,7 @@ xlabel={$\rho$}, ylabel={Avg. \# iter.}, ymode=log, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[ForestGreen, line width=1pt] table [col sep=comma, x=rho, y=k_avg, @@ -468,10 +477,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$ - \cite[\text{PEGReg252x504}]{mackay_enc}} + \citereference{Mac23, PEGReg252x504}} \end{subfigure}% \end{figure} \end{minipage}% @@ -500,5 +509,10 @@ \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} diff --git a/latex/presentations/final/sections/comparison.tex b/latex/presentations/final/sections/comparison.tex index 71667b8..5aa3f4b 100644 --- a/latex/presentations/final/sections/comparison.tex +++ b/latex/presentations/final/sections/comparison.tex @@ -11,13 +11,15 @@ \begin{frame}[t] \frametitle{Comparison: Convergence Behavior} + \vspace*{-3mm} + \begin{itemize} - \item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} + \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484} \end{itemize} \begin{figure}[H] \centering - + \begin{subfigure}[t]{0.5\textwidth} \centering @@ -105,6 +107,14 @@ \begin{itemize} \item Minimum number of iterations independant of SNR \end{itemize} + + \bigskip + \smallskip + \smallskip + \smallskip + + \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -114,11 +124,11 @@ \vspace*{-6.5mm} \begin{itemize} - \item The points shown are from the following codes: BCH $\left( 31, 11 \right)$; - BCH $\left( 31, 26 \right)$; \\ - \cite[\text{96.3.965; 204.33.484; 204.55.187; 408.33.844; PEGReg252x504}]{mackay_enc} + \item Codes: BCH $\left( 31, 11 \right)$; BCH $\left( 31, 26 \right)$; + \citereference{Mac23, 96.3.965; 204.33.484; 204.55.187; 408.33.844; PEGReg252x504} \item Measured performance: $\sim\SI{10000}{}$ frames/s on Intel Core i7-7700HQ @ 2.80GHz; $n=204$ + \item Both algorithms are $\mathcal{O}\left( n \right)$ on average \end{itemize} \begin{figure}[H] @@ -130,8 +140,8 @@ xlabel={$n$}, ylabel={time per frame (s)}, legend style={at={(0.05,0.6)},anchor=south west}, legend cell align={left}, - width=0.45\textwidth, - height=0.3375\textwidth, + width=0.43\textwidth, + height=0.32\textwidth, ] \addplot[RedOrange, only marks, mark=square*] table [col sep=comma, x=n, y=spf] @@ -151,10 +161,12 @@ \end{tikzpicture} \end{figure} - \begin{itemize} - \item Both algorithms are $\mathcal{O}\left( n \right)$ on average - \item LP decoding implementation significantly faster - \end{itemize} + \bigskip + \smallskip + \smallskip + + \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} @@ -165,7 +177,7 @@ \vspace{-0.5cm} \begin{itemize} - \item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} + \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484} \end{itemize} \begin{figure}[H] @@ -179,8 +191,8 @@ legend columns=2, legend style={at={(0.5,-0.45)},anchor=south}, ymax=1.5, ymin=3e-8, - width=0.48\textwidth, - height=0.35\textwidth, + width=0.45\textwidth, + height=0.33\textwidth, ] \addplot[RedOrange, mark=*, line width=1pt] table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}] @@ -208,8 +220,8 @@ legend columns=2, legend style={at={(0.5,-0.45)},anchor=south}, ymax=1.5, ymin=3e-8, - width=0.48\textwidth, - height=0.35\textwidth, + width=0.45\textwidth, + height=0.33\textwidth, ] \addplot[RedOrange, mark=*, solid, line width=1pt] @@ -256,6 +268,13 @@ \end{axis} \end{tikzpicture} \end{figure} + + \smallskip + \smallskip + \smallskip + + \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -318,7 +337,7 @@ \centering \begin{figure}[H] - \vspace*{-0.7cm} + \vspace*{-0.8cm} \centering \begin{subfigure}[t]{0.33\textwidth} @@ -331,7 +350,7 @@ ymode=log, ymax=1.5, ymin=8e-5, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[RedOrange, line width=1pt, mark=*, solid] @@ -351,10 +370,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$ - \cite[\text{96.3.965}]{mackay_enc}} + \citereference{Mac23, 96.33.965}} \end{subfigure}% \begin{subfigure}[t]{0.33\textwidth} \centering @@ -366,7 +385,7 @@ ymode=log, ymax=1.5, ymin=8e-5, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[RedOrange, line width=1pt, mark=*] @@ -390,10 +409,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$ - \cite[\text{204.33.484}]{mackay_enc}} + \citereference{Mac23, 204.33.484}} \end{subfigure}% \begin{subfigure}[t]{0.33\textwidth} \centering @@ -405,7 +424,7 @@ ymode=log, ymax=1.5, ymin=8e-5, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[RedOrange, line width=1pt, mark=*] @@ -421,12 +440,14 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$ - \cite[\text{408.33.844}]{mackay_enc}} + \citereference{Mac23, 408.33.844}} \end{subfigure}% + \vspace*{-2mm} + \begin{subfigure}[t]{0.33\textwidth} \centering @@ -437,7 +458,7 @@ ymode=log, ymax=1.5, ymin=8e-5, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[RedOrange, line width=1pt, mark=*] @@ -459,7 +480,7 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{BCH code with $n=31, k=26$} \end{subfigure}% @@ -473,7 +494,7 @@ ymode=log, ymax=1.5, ymin=8e-5, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[RedOrange, line width=1pt, mark=*] @@ -488,10 +509,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$ - \cite[\text{204.55.187}]{mackay_enc}} + \citereference{Mac23, 204.55.187}} \end{subfigure}% \begin{subfigure}[t]{0.33\textwidth} \centering @@ -503,7 +524,7 @@ ymode=log, ymax=1.5, ymin=8e-5, width=1.2\textwidth, - height=0.85\textwidth, + height=0.825\textwidth, ] \addplot[RedOrange, line width=1pt, mark=*] @@ -519,10 +540,10 @@ \end{axis} \end{tikzpicture} - \vspace*{-1mm} + \vspace*{-2mm} \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$ - \cite[\text{PEGReg252x504}]{mackay_enc}} + \citereference{Mac23, PEGReg252x504}} \end{subfigure}% \end{figure} \end{minipage}% @@ -548,19 +569,17 @@ \addlegendimage{NavyBlue, line width=1pt, mark=*} \addlegendentry{ADMM} -% \addlegendimage{RoyalPurple, line width=1pt, mark=*} -% \addlegendentry{BP} - \addlegendimage{Black, line width=1pt, mark=*} \addlegendentry{ML} - -% \addlegendimage{PineGreen, line width=1pt, mark=triangle*, solid} -% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -700,24 +719,3 @@ return $\tilde{\boldsymbol{c}}$ \end{figure}% \end{frame} -\begin{frame}[t] - \frametitle{Conclusion} - - \begin{itemize} - \item Analysis of the general behavior of the two decoding algorithms - \begin{itemize} - \item Parameter choice based on decoding performance and time complexity - \item Verification of theoretical considerations with simulation results - \end{itemize} - \item Suggestion for improvement of proximal decoding - \begin{itemize} - \item Addition of "ML-in-the-List" step - \item Up to $\sim \SI{1}{dB}$ gain under certain conditions - \end{itemize} - \item Comparison - \begin{itemize} - \item based on simulation results - \item based on their theoretical structure - \end{itemize} - \end{itemize} -\end{frame} diff --git a/latex/presentations/final/sections/conclusion.tex b/latex/presentations/final/sections/conclusion.tex new file mode 100644 index 0000000..c63f302 --- /dev/null +++ b/latex/presentations/final/sections/conclusion.tex @@ -0,0 +1,26 @@ +\section{Conclusion and Outlook}% +\label{sec:Comparison} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}[t] + \frametitle{Conclusion} + + \begin{itemize} + \item Analysis of the general behavior of the two decoding algorithms + \begin{itemize} + \item Parameter choice based on decoding performance and time complexity + \item Verification of theoretical considerations with simulation results + \end{itemize} + \item Suggestion for improvement of proximal decoding + \begin{itemize} + \item Addition of "ML-in-the-List" step + \item Up to $\sim \SI{1}{dB}$ gain under certain conditions + \end{itemize} + \item Comparison + \begin{itemize} + \item based on simulation results + \item based on their theoretical structure + \end{itemize} + \end{itemize} +\end{frame} diff --git a/latex/presentations/final/sections/lp_dec_using_admm.tex b/latex/presentations/final/sections/lp_dec_using_admm.tex index 2a18d2d..9c988d0 100644 --- a/latex/presentations/final/sections/lp_dec_using_admm.tex +++ b/latex/presentations/final/sections/lp_dec_using_admm.tex @@ -7,7 +7,9 @@ \label{sub:LP Decoding using ADMM} \begin{frame}[t] - \frametitle{LP Decoding \cite{feldman_paper}} + \frametitle{LP Decoding \citereference{FWK05}} + + \vspace*{-3mm} \begin{itemize} \item General reformulation of ML decoding as a linear program (LP) @@ -36,6 +38,13 @@ \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} @@ -656,12 +665,16 @@ \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}} + \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} @@ -692,6 +705,12 @@ % - \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} @@ -703,12 +722,14 @@ \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 $\left( \boldsymbol{u}_j \right)_i$}: + \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*{-3mm} \begin{alignat*}{3} \tilde{c}_i &\leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( @@ -726,12 +747,31 @@ - \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 \cite{original_admm}, \cite{efficient_lp_dec_admm}, - \cite{lautern} - \item The approach chosen here is the one described in \cite{original_admm} + \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 + \smallskip + + \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} @@ -827,9 +867,11 @@ \begin{frame}[t] \frametitle{LP Decoding using ADMM: Frame Error Rate} + \vspace*{-6mm} + \begin{itemize} - \item ``Margulis'' LDPC code with $n = 2640$, $k = 1320$ - \cite[\text{Margulis2640.1320.3}]{mackay_enc} + \item ``Margulis'' LDPC code with $n = 2640$, $k = 1320$ + \citereference{Mac23, Margulis2640.1320.3} % ; $K=200, \mu = 3.3, \rho=1.9, % \epsilon_{\text{pri}} = 10^{-5}, \epsilon_{\text{dual}} = 10^{-5}$ \end{itemize} @@ -852,50 +894,7 @@ discard if gt={SNR}{2.2}, ] {res/admm/fer_paper_margulis.csv}; - \addlegendentry{ADMM (Barman et al.)} - \addplot[RoyalPurple, line width=1pt, mark=*] - table [col sep=comma, x=SNR, y=FER, discard if gt={SNR}{2.2},] - {res/generic/fer_bp_mackay_margulis.csv}; - \addlegendentry{BP (Barman et al.)} - \end{axis} - \end{tikzpicture} - \end{figure} - - \begin{itemize} - \item Comparison of simulation with results from Barman et al. \cite{original_admm} - \end{itemize} - -\end{frame} - -\begin{frame}[t] - \frametitle{LP Decoding using ADMM: Frame Error Rate} - - \begin{itemize} - \item ``Margulis'' LDPC code with $n = 2640$, $k = 1320$ - \cite[\text{Margulis2640.1320.3}]{mackay_enc} -% ; $K=200, \mu = 3.3, \rho=1.9, -% \epsilon_{\text{pri}} = 10^{-5}, \epsilon_{\text{dual}} = 10^{-5}$ - \end{itemize} - - \begin{figure}[H] - \centering - - \begin{tikzpicture} - \begin{axis}[ - grid=both, - xlabel={$E_b / N_0 \left( \text{dB} \right) $}, ylabel={FER}, - ymode=log, - width=0.45\textwidth, - height=0.325\textwidth, - %legend style={at={(0.03,0.04)},anchor=south west}, - legend pos=outer north east, - ] - \addplot[Emerald, line width=1pt, mark=*] - table [col sep=comma, x=SNR, y=FER, - discard if gt={SNR}{2.2}, - ] - {res/admm/fer_paper_margulis.csv}; - \addlegendentry{ADMM (Barman et al.)} + \addlegendentry{ADMM \citereference{$\text{Bar}^+\text{13}$}} \addplot[NavyBlue, mark=*, line width=1pt, densely dashed] table [col sep=comma, x=SNR, y=FER,] {res/admm/ber_margulis264013203.csv}; @@ -903,14 +902,25 @@ \addplot[RoyalPurple, line width=1pt, mark=*] table [col sep=comma, x=SNR, y=FER, discard if gt={SNR}{2.2},] {res/generic/fer_bp_mackay_margulis.csv}; - \addlegendentry{BP (Barman et al.)} + \addlegendentry{BP \citereference{$\text{Bar}^+\text{13}$}} \end{axis} \end{tikzpicture} \end{figure}% \begin{itemize} - \item Comparison of simulation with results from Barman et al. \cite{original_admm} + \item Comparison of simulation with results from Barman et al. + \citereference{$\text{Bar}^+\text{13}$} \end{itemize} + + \bigskip + + \addreferences + {Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}. + Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.} + {$\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.} + \stopreferences \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -918,8 +928,10 @@ \frametitle{LP Decoding using ADMM: Choice of Penalty\\ Parameters} + \vspace*{-5mm} + \begin{itemize} - \item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} + \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484} \end{itemize} \begin{figure}[H] @@ -933,8 +945,8 @@ grid=both, xlabel={$\mu$}, ylabel={FER}, ymode=log, - width=0.9\textwidth, - height=0.675\textwidth, + width=0.8\textwidth, + height=0.6\textwidth, ] \addplot[ForestGreen, line width=1pt, densely dashed, mark=*] table [col sep=comma, x=mu, y=FER, @@ -959,8 +971,8 @@ grid=both, xlabel={$\rho$}, ylabel={FER}, ymode=log, - width=0.9\textwidth, - height=0.675\textwidth, + width=0.8\textwidth, + height=0.6\textwidth, ] \addplot[ForestGreen, line width=1pt, densely dashed, mark=*] table [col sep=comma, x=rho, y=FER, @@ -1003,15 +1015,22 @@ \item Similar to $\gamma$ with proximal decoding: no clear optimum \end{itemize} + \bigskip + \bigskip + + \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[t] \frametitle{LP Decoding using ADMM: Choice of Penalty\\ Parameters} - + + \vspace*{-5mm} + \begin{itemize} - \item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} + \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484} \end{itemize} \begin{figure}[H] @@ -1024,8 +1043,8 @@ \begin{axis}[ grid=both, xlabel={$\mu$}, ylabel={Average \# of iterations}, - width=0.9\textwidth, - height=0.675\textwidth, + width=0.8\textwidth, + height=0.6\textwidth, ] \addplot[ForestGreen, line width=1pt, densely dashed, mark=*] table [col sep=comma, x=mu, y=k_avg, @@ -1049,8 +1068,8 @@ \begin{axis}[ grid=both, xlabel={$\rho$}, ylabel={Average \# of iterations}, - width=0.9\textwidth, - height=0.675\textwidth, + width=0.8\textwidth, + height=0.6\textwidth, ] \addplot[ForestGreen, line width=1pt, densely dashed, mark=*] table [col sep=comma, x=rho, y=k_avg, @@ -1092,5 +1111,13 @@ \begin{itemize} \item For lower decoding time, choose low $\mu$ and high $\rho$ \end{itemize} + + \bigskip + \smallskip + \smallskip + \smallskip + + \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} diff --git a/latex/presentations/final/sections/proximal_decoding.tex b/latex/presentations/final/sections/proximal_decoding.tex index 100d08c..f4e12bc 100644 --- a/latex/presentations/final/sections/proximal_decoding.tex +++ b/latex/presentations/final/sections/proximal_decoding.tex @@ -7,9 +7,9 @@ \label{sub:Decoding Algorithm} \begin{frame}[t] - \frametitle{Proximal Decoding: General Idea \cite{proximal_paper}} + \frametitle{Proximal Decoding: General Idea \citereference{WT22}} - \vspace*{-0.3cm} + \vspace*{-0.7cm} \begin{itemize} \item MAP rule as continuous maximization problem: @@ -59,6 +59,13 @@ \end{minipage} \hfill \end{itemize} + + \smallskip + \smallskip + + \addreference{WT22}{Tadashi Wadayama; Satoshi Takabe: \emph{Proximal Decoding for LDPC + Codes}. IEICE Transactions on Fundamentals of Electronics, Communications and Computer + Sciences advpub (2022), 2022TAP0002.} \end{frame} @@ -66,6 +73,8 @@ \begin{frame}[t] \frametitle{Proximal Decoding: General Idea} + \vspace*{-3mm} + \begin{itemize} \item Objective function: \begin{align*} @@ -74,7 +83,7 @@ + \gamma h\left( \tilde{\boldsymbol{x}} \right) \end{align*} \note{Notational difference between $f$ and $f_X$ or $f_Y$} - \item Proximal operator \cite{proximal_algorithms}: + \item Proximal operator \citereference{PB14}: \begin{align*} \text{prox}_{\gamma h} \left( \tilde{\boldsymbol{x}} \right) &\equiv \argmin_{\boldsymbol{t}\in\mathbb{R}^n} @@ -96,6 +105,11 @@ \hspace{10mm} \text{``Code proximal step''} \end{align*} \end{itemize} + + \bigskip + + \addreference{PB14}{Neal Parikh; Stephen Boyd: \emph{Proximal Algorithms}. + Found. Trends Optim. 1.3 (Jan. 2014), pp. 127–239.} \end{frame} @@ -103,7 +117,7 @@ \begin{frame}[t, fragile] \frametitle{Proximal Decoding: Algorithm} \begin{itemize} - \item Iterative decoding algorithm \cite{proximal_paper}: + \item Iterative decoding algorithm: \end{itemize} \vspace{2mm} @@ -130,8 +144,10 @@ return $\boldsymbol{\hat{c}}$ \begin{frame}[t] \frametitle{Proximal Decoding: Bit Error Rate} + \vspace*{-7mm} + \begin{itemize} - \item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} + \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484} \end{itemize} \begin{figure}[H] @@ -159,7 +175,7 @@ return $\boldsymbol{\hat{c}}$ \addlegendentry{$\gamma = 0.15$} \addplot [ForestGreen, mark=*, line width=1pt] table [x=SNR, y=gamma_0_15, col sep=comma] {res/ber_paper.csv}; - \addlegendentry{$\gamma = 0.15$ (Wadayama et al.)} + \addlegendentry{$\gamma = 0.15$ \citereference{WT22}} \addplot [NavyBlue, mark=triangle, dashed, line width=1pt] table [x=SNR, y=BER, col sep=comma, @@ -169,7 +185,7 @@ return $\boldsymbol{\hat{c}}$ \addlegendentry{$\gamma = 0.01$} \addplot [NavyBlue, mark=*, line width=1pt] table [x=SNR, y=gamma_0_01, col sep=comma] {res/ber_paper.csv}; - \addlegendentry{$\gamma = 0.01$ (Wadayama et al.)} + \addlegendentry{$\gamma = 0.01$ \citereference{WT22}} \addplot [RedOrange, mark=triangle, dashed, line width=1pt] table [x=SNR, y=BER, col sep=comma, @@ -179,19 +195,29 @@ return $\boldsymbol{\hat{c}}$ \addlegendentry{$\gamma = 0.05$} \addplot [RedOrange, mark=*, line width=1pt] table [x=SNR, y=gamma_0_05, col sep=comma] {res/ber_paper.csv}; - \addlegendentry{$\gamma = 0.05$ (Wadayama et al.)} + \addlegendentry{$\gamma = 0.05$ \citereference{WT22}} \addplot [RoyalPurple, mark=*, line width=1pt] table [x=SNR, y=BP, col sep=comma] {res/ber_paper.csv}; - \addlegendentry{BP} + \addlegendentry{BP \citereference{WT22}} \end{axis} \end{tikzpicture} \end{figure} + + \vspace*{-2mm} \begin{itemize} - \item Comparison of simulation with results of Wadayama et al. \cite{proximal_paper} + \item Comparison of simulation with results of Wadayama et al. \citereference{WT22} \end{itemize} + + \bigskip + + \addreference{WT22}{Tadashi Wadayama; Satoshi Takabe: \emph{Proximal Decoding for LDPC + Codes}. IEICE Transactions on Fundamentals of Electronics, Communications and Computer + Sciences advpub (2022), 2022TAP0002.} + \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} @@ -199,8 +225,10 @@ return $\boldsymbol{\hat{c}}$ \begin{frame}[t] \frametitle{Proximal Decoding: Choice of $\gamma$} + \vspace*{-5mm} + \begin{itemize} - \item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} + \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484} \end{itemize} \begin{figure}[H] @@ -273,6 +301,12 @@ return $\boldsymbol{\hat{c}}$ \begin{itemize} \item Not great benefit in finding the optimal value for $\gamma$ \end{itemize} + + \bigskip + + \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} @@ -495,12 +529,14 @@ return $\boldsymbol{\hat{c}}$ \begin{frame}[t, fragile] \frametitle{Proximal Decoding: Frame Error Rate} + \vspace*{-3mm} + \begin{itemize} \item (3,6) regular LDPC code with $n=204$,\\ - $k=102$ \cite[\text{204.33.484}]{mackay_enc} + $k=102$ \citereference{Mac23, 204.33.484} \end{itemize} - \vspace*{-5mm} + \vspace*{-2mm} \begin{minipage}{.4\textwidth} \centering @@ -521,6 +557,11 @@ end for return $\boldsymbol{\hat{c}}$ \end{algorithm} \end{figure} + + \vspace*{-4mm} + \hspace*{-4mm} + \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{minipage}% \begin{minipage}{.6\textwidth} \centering @@ -854,8 +895,8 @@ return $\boldsymbol{\hat{c}}$ \end{axis} \end{tikzpicture} - \caption{$\nabla L \left(\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)$ - for a repetition code with $n=2$\footnotemark} + \caption{$\nabla L \left(\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)$% + \footnotemark{} for a repetition code with $n=2$} \end{subfigure}% \begin{subfigure}[c]{0.5\textwidth} \centering @@ -894,9 +935,7 @@ return $\boldsymbol{\hat{c}}$ \footnotetext{In an AWGN Channel $\nabla L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}}\right) - \propto \left( \tilde{\boldsymbol{x}} - \boldsymbol{y} \right)$ - \cite[Sec. 4.1]{proximal_paper}} - + \propto \left( \tilde{\boldsymbol{x}} - \boldsymbol{y} \right)$} \end{frame} @@ -904,9 +943,11 @@ return $\boldsymbol{\hat{c}}$ \begin{frame}[t] \frametitle{Proximal Decoder: Oscillation of $\nabla h\left( \tilde{\boldsymbol{x}} \right) $} + \vspace*{-3mm} + \begin{itemize} \item Single decoding using a (3,6) regular LDPC code with $n=204, k=102$ - \cite[\text{204.33.484}]{mackay_enc} + \citereference{Mac23, 204.33.484} % ; $\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$ \end{itemize} @@ -969,6 +1010,12 @@ return $\boldsymbol{\hat{c}}$ \item For larger $n$, the gradient itself starts to oscillate \item Dynamic range of oscillation highly correlated with probability of bit error \end{itemize} + + \bigskip + \bigskip + + \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} @@ -1086,7 +1133,7 @@ return $\boldsymbol{\hat{c}}$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[t, fragile] - \frametitle{Proximal Decoding: Improvement using \\``ML-in-the-List''} + \frametitle{Proximal Decoding: Improvement} \vspace*{-0.5cm} @@ -1147,12 +1194,12 @@ $\textcolor{KITblue}{\textbf{return }\boldsymbol{\hat{c}}_l\text{ with lowest }d %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[t] - \frametitle{Proximal Decoding: Improvement using \\``ML-in-the-List''} + \frametitle{Proximal Decoding: Improvement} - \vspace{-0.5cm} + \vspace{-0.6cm} \begin{itemize} - \item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} + \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484} \end{itemize} \begin{figure}[H] @@ -1166,8 +1213,8 @@ $\textcolor{KITblue}{\textbf{return }\boldsymbol{\hat{c}}_l\text{ with lowest }d legend columns=2, legend style={at={(0.5,-0.45)},anchor=south}, ymax=1.5, ymin=3e-8, - width=0.48\textwidth, - height=0.33\textwidth, + width=0.45\textwidth, + height=0.3\textwidth, ] \addplot[ForestGreen, mark=*, line width=1pt] table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.15}] @@ -1199,8 +1246,8 @@ $\textcolor{KITblue}{\textbf{return }\boldsymbol{\hat{c}}_l\text{ with lowest }d legend columns=2, legend style={at={(0.5,-0.45)},anchor=south}, ymax=1.5, ymin=3e-8, - width=0.48\textwidth, - height=0.33\textwidth, + width=0.45\textwidth, + height=0.3\textwidth, ] \addplot[ForestGreen, mark=*, solid, line width=1pt] @@ -1256,6 +1303,11 @@ $\textcolor{KITblue}{\textbf{return }\boldsymbol{\hat{c}}_l\text{ with lowest }d \begin{itemize} \item Up to $\sim \SI{1}{dB}$ improvement \end{itemize} + + \bigskip + + \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} diff --git a/latex/presentations/final/sections/theoretical_background.tex b/latex/presentations/final/sections/theoretical_background.tex index 9a88f90..6f42622 100644 --- a/latex/presentations/final/sections/theoretical_background.tex +++ b/latex/presentations/final/sections/theoretical_background.tex @@ -9,18 +9,37 @@ \begin{frame}[t] \frametitle{Motivation} \begin{itemize} - \item The general ML decoding problem is NP-complete \cite{ml_np_hard_proof} - \item The iterative message–passing algorithms preferred in practice do not guarantee - optimality when the graph contains cycles \cite{ldpc_conv} - \item The standard message-passing algorithms are often difficult to - analyze \cite{feldman_thesis} + \item General ML decoding problem NP-complete \citereference{BMT78} + \item Iterative message–passing algorithms preferred in practice do not guarantee + optimality when the graph contains cycles \citereference{KTP19} + \item Standard message-passing algorithms difficult to analyze \citereference{FEL03} \end{itemize} + + \vspace{3.5cm} + + \addreferences + {BMT78}{E. Berlekamp; R. McEliece; H. van Tilborg: \emph{On the inherent intractability + of certain coding problems (Corresp.)}. + IEEE Transactions on Information Theory 24.3 (May 1978), pp. 384–386.} + {KTP19}{Banu Kabakulak; Z. Caner Taşkın; Ali Emre Pusane: \emph{Optimization–based + decoding algorithms for LDPC convolutional codes in communication systems}. + IISE Transactions 51.10 (2019), pp. 1061–1074.} + {FEL03}{Jon Feldman: \emph{Decoding error-correcting codes via linear programming}. + PhD thesis. MIT, 2003.} + \stopreferences \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[t] \frametitle{Previous Work} + + \vspace*{-5mm} + + \begin{itemize} + \item Bachelor's thesis by Yanixa Lu \citereference{Lu22} + \end{itemize} + \begin{figure}[h] \centering @@ -39,14 +58,17 @@ \fbox{\includegraphics[page=60,width=.6\textwidth]{res/Bachelor_Thesis_Yanxia_Lu}} \end{subfigure}% - - \caption{Bachelor's Thesis by Yanxia Lu \cite{yanxia_lu_thesis}} \end{figure} \begin{itemize} \item Analysis of ``Proximal Decoding'' \item Analysis of ``Interior Point Decoding'' - \end{itemize} + \end{itemize} + + \bigskip + + \addreference{Lu22}{Yanxia Lu: \emph{Realization of Channel Decoding Using Optimization + Techniques}. Bachelor’s Thesis. KIT, 2022.} \end{frame}