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Added 6-sublot BER surface plot; Added pgf externalization

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Andreas Tsouchlos 2023-01-24 00:07:15 +01:00
parent 87ec681ae9
commit 77fce73273
4 changed files with 391 additions and 98 deletions
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4
latex/presentations/midterm/presentation.bib
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@ -25,7 +25,7 @@
journal = {IEEE Transactions on Information Theory},
author = {Berlekamp, E. and McEliece, R. and van Tilborg, H.},
year = {1978},
month = {May},
month = {5},
pages = {384–386},
% url = {https://authors.library.caltech.edu/5607/1/BERieeetit78.pdf}
}
@ -64,7 +64,7 @@
url = {https://doi.org/10.1561/2400000003},
doi = {10.1561/2400000003},
journal = {Found. Trends Optim.},
month = {jan},
month = {1},
pages = {127–239},
numpages = {113}
}

2
latex/presentations/midterm/presentation.tex
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@ -23,6 +23,8 @@
%\setbeamertemplate{note page}[plain]
%\setbeameroption{show notes on second screen=right}
\usepgfplotslibrary{colorbrewer}
\usetikzlibrary{external}
\tikzexternalize[prefix=build/]
\usepackage{csquotes}
\usepackage[citestyle=numeric, style=alphabetic, backend=biber,

2
latex/presentations/midterm/sections/decoding_algorithms.tex
View File

@ -83,7 +83,7 @@
\begin{frame}[t, fragile]
\frametitle{Proximal Decoding: Algorithm}
\begin{itemize}
\item Resulting terative decoding algorithm:
\item Resulting terative decoding algorithm \cite{proximal_paper}:
\vspace{2mm}
\begin{algorithm}[caption={}, label={}]
$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$

481
latex/presentations/midterm/sections/examination_results.tex
View File

@ -95,7 +95,9 @@
legend style={at={(0.05,0.05)},anchor=south west},
ymin=3e-7, ymax=1.5,]
\foreach \gamma in {0.01, 0.05, 0.15}{
\addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{\gamma}] {res/2d_ber_fer_dfr_20433484_proximal.csv};
\addplot table [x=SNR, y=BER,
col sep=comma, discard if not={gamma}{\gamma}]
{res/2d_ber_fer_dfr_20433484_proximal.csv};
\legend{\gamma}
}
\legend{$\gamma=0.01$, $\gamma=0.05$, $\gamma=0.15$}
@ -111,13 +113,26 @@
xlabel={SNR},
ylabel={$\gamma$},
zlabel={BER},]
\addplot3[surf, mesh/rows=17, mesh/cols=14, colormap/viridis] table [col sep=comma, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv};
\addplot3[surf,
mesh/rows=17, mesh/cols=14,
colormap/viridis] table [col sep=comma,
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20433484_proximal.csv};
\addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
\addplot3[red, line width=1.5] table[col sep=comma, discard if not={gamma}{0.05}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv};
\addplot3[red, line width=1.5] table [col sep=comma,
discard if not={gamma}{0.05},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20433484_proximal.csv};
\addlegendentry{$\gamma = 0.05$}
\addplot3[blue, line width=1.5] table[col sep=comma, discard if not={gamma}{0.01}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv};
\addplot3[blue, line width=1.5] table [col sep=comma,
discard if not={gamma}{0.01},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20433484_proximal.csv};
\addlegendentry{$\gamma = 0.01$}
\addplot3[brown, line width=1.5] table[col sep=comma, discard if not={gamma}{0.15}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv};
\addplot3[brown, line width=1.5] table [col sep=comma,
discard if not={gamma}{0.15},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20433484_proximal.csv};
\addlegendentry{$\gamma = 0.15$}
\end{axis}
\end{tikzpicture}
@ -134,6 +149,215 @@
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{Proximal Decoding: Choice of $\gamma$}
\begin{figure}[H]
\vspace*{-0.5cm}
\centering
\begin{subfigure}[c]{0.33\textwidth}
\centering
\begin{tikzpicture}[scale=0.35]
\begin{axis}[view={75}{60},
zmode=log,
xlabel={SNR},
ylabel={$\gamma$},
zlabel={BER},]
\addplot3[surf,
mesh/rows=17, mesh/cols=10,
colormap/viridis] table [col sep=comma,
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_963965.csv};
\addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
\addplot3[red, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.05},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_963965.csv};
\addlegendentry{$\gamma = 0.05$}
\addplot3[blue, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.01},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_963965.csv};
\addlegendentry{$\gamma = 0.01$}
\addplot3[brown, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.15},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_963965.csv};
\addlegendentry{$\gamma = 0.15$}
\end{axis}
\end{tikzpicture}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$\\ \cite[Code: 96.3.965]{mackay_enc}}
\end{subfigure}%
\begin{subfigure}[c]{0.33\textwidth}
\centering
\begin{tikzpicture}[scale=0.35]
\begin{axis}[view={75}{60},
zmode=log,
xlabel={SNR},
ylabel={$\gamma$},
zlabel={BER},]
\addplot3[surf,
mesh/rows=17, mesh/cols=14,
colormap/viridis] table [col sep=comma,
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20433484.csv};
\addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
\addplot3[red, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.05},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20433484.csv};
\addlegendentry{$\gamma = 0.05$}
\addplot3[blue, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.01},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20433484.csv};
\addlegendentry{$\gamma = 0.01$}
\addplot3[brown, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.15},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20433484.csv};
\addlegendentry{$\gamma = 0.15$}
\end{axis}
\end{tikzpicture}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$\\ \cite[Code: 204.33.484]{mackay_enc}}
\end{subfigure}%
\begin{subfigure}[c]{0.33\textwidth}
\centering
\begin{tikzpicture}[scale=0.35]
\begin{axis}[view={75}{60},
zmode=log,
xlabel={SNR},
ylabel={$\gamma$},
zlabel={BER},]
\addplot3[surf,
mesh/rows=17, mesh/cols=10,
colormap/viridis] table [col sep=comma,
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_40833844.csv};
\addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
\addplot3[red, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.05},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_40833844.csv};
\addlegendentry{$\gamma = 0.05$}
\addplot3[blue, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.01},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_40833844.csv};
\addlegendentry{$\gamma = 0.01$}
\addplot3[brown, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.15},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_40833844.csv};
\addlegendentry{$\gamma = 0.15$}
\end{axis}
\end{tikzpicture}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$\\ \cite[Code: 408.33.844]{mackay_enc}}
\end{subfigure}
\begin{subfigure}[c]{0.33\textwidth}
\centering
\begin{tikzpicture}[scale=0.35]
\begin{axis}[view={75}{60},
zmode=log,
xlabel={SNR},
ylabel={$\gamma$},
zlabel={BER},]
\addplot3[surf,
mesh/rows=17, mesh/cols=10,
colormap/viridis] table [col sep=comma,
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_bch_31_26.csv};
\addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
\addplot3[red, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.05},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_bch_31_26.csv};
\addlegendentry{$\gamma = 0.05$}
\addplot3[blue, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.01},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_bch_31_26.csv};
\addlegendentry{$\gamma = 0.01$}
\addplot3[brown, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.15},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_bch_31_26.csv};
\addlegendentry{$\gamma = 0.15$}
\end{axis}
\end{tikzpicture}
\caption{BCH code with $n=31, k=26$}
\end{subfigure}%
\begin{subfigure}[c]{0.33\textwidth}
\centering
\begin{tikzpicture}[scale=0.35]
\begin{axis}[view={75}{60},
zmode=log,
xlabel={SNR},
ylabel={$\gamma$},
zlabel={BER},]
\addplot3[surf,
mesh/rows=17, mesh/cols=10,
colormap/viridis] table [col sep=comma,
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20455187.csv};
\addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
\addplot3[red, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.05},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20455187.csv};
\addlegendentry{$\gamma = 0.05$}
\addplot3[blue, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.01},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20455187.csv};
\addlegendentry{$\gamma = 0.01$}
\addplot3[brown, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.15},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_20455187.csv};
\addlegendentry{$\gamma = 0.15$}
\end{axis}
\end{tikzpicture}
\caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$\\ \cite[Code: 204.55.187]{mackay_enc}}
\end{subfigure}%
\begin{subfigure}[c]{0.33\textwidth}
\centering
\begin{tikzpicture}[scale=0.35]
\begin{axis}[view={75}{60},
zmode=log,
xlabel={SNR},
ylabel={$\gamma$},
zlabel={BER},]
\addplot3[surf,
mesh/rows=17, mesh/cols=10,
colormap/viridis] table [col sep=comma,
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_pegreg252x504.csv};
\addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
\addplot3[red, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.05},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_pegreg252x504.csv};
\addlegendentry{$\gamma = 0.05$}
\addplot3[blue, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.01},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_pegreg252x504.csv};
\addlegendentry{$\gamma = 0.01$}
\addplot3[brown, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.15},
x=SNR, y=gamma, z=BER]
{res/2d_ber_fer_dfr_pegreg252x504.csv};
\addlegendentry{$\gamma = 0.15$}
\end{axis}
\end{tikzpicture}
\caption{LDPC code (Progressive Edge Growth Construction) with $n=504, k=252$\\ \cite[Code: PEGReg252x504]{mackay_enc}}
\end{subfigure}%
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t, fragile]
\frametitle{Proximal Decoding: Frame Error Rate}
@ -149,7 +373,10 @@
\begin{minipage}{.4\textwidth}
\centering
\begin{algorithm}[caption={}, label={},
\begin{figure}[htpb]
\centering
\begin{algorithm}[caption={}, label={},
basicstyle=\fontsize{7.5}{9.5}\selectfont
]
$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$
@ -161,7 +388,10 @@ for $k=0$ to $K-1$ do
If $\boldsymbol{\hat{x}}$ passes the parity check condition, break the loop.
end for
Output $\boldsymbol{\hat{x}}$
\end{algorithm}
\end{algorithm}
\caption{Proximal decoding algorithm \cite{proximal_paper}}
\end{figure}
\end{minipage}%
\begin{minipage}{.6\textwidth}
\centering
@ -234,17 +464,118 @@ Output $\boldsymbol{\hat{x}}$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\tikzbracemark}[1]{\tikz[overlay,remember picture] \node (#1) {};}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newcommand{\tikzbracemark}[1]{\tikz[overlay,remember picture] \node (#1) {};}
%
%\newcommand*{\AddNote}[4]{%
% \begin{tikzpicture}[overlay, remember picture]
% \draw [decoration={brace,amplitude=0.5em},decorate,ultra thick]
% ($(#3)!([yshift=1.5ex]#1)!($(#3)-(0,1)$)$) --
% ($(#3)!(#2)!($(#3)-(0,1)$)$)
% node [align=center, text width=2cm, pos=0.5, anchor=west] {#4};
% \end{tikzpicture}
%}%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[t, fragile]
% \frametitle{Proximal Decoding: Improvement using ``ML-on-List''}
% \setcounter{footnote}{0}
%
% \begin{itemize}
% \item Comparison of proximal \& hybrid-proximal-ML\\
% decoding simulation
% \footnote{(3,6) regular LDPC Code with $n=204, k=102$
% \cite[Code: 204.33.484]{mackay_enc}}
% results
% \end{itemize}
%
% \begin{minipage}{.4\textwidth}
% \centering
%
% \begin{algorithm}[caption={}, label={},
% basicstyle=\fontsize{6.5}{7.5}\selectfont
% ]
%$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$$\hspace{4.185cm}\tikzbracemark{prox-start}$
%for $k=0$ to $K-1$ do
% $\boldsymbol{r}^{\left( k+1 \right)} = \boldsymbol{s}^{(k)} - \omega \nabla L \left( \boldsymbol{s}^{(k)}; \boldsymbol{y} \right) $
% Compute $\nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)$
% $\boldsymbol{s}^{\left( k+1 \right)} = \boldsymbol{r}^{(k+1)} - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right) $
% $\boldsymbol{\hat{x}} = \text{sign}\left( \boldsymbol{s}^{\left( k+1 \right) } \right) $
% If $\boldsymbol{\hat{x}}$ passes the parity check condition, output $\boldsymbol{\hat{x}}$
%end for $\tikzbracemark{prox-end}$
%Find $N$ most probably wrong bits $\hspace{2cm}\tikzbracemark{ml-start}$
%Generate variations $\boldsymbol{\tilde{x}}_n$ of $\boldsymbol{\hat{x}}$ with the $N$ bits modified
%Compute $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right) \forall n \in \left[ 1 : N-1 \right] $
%Output $\boldsymbol{\tilde{x}}_n$ with lowest $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)$ $\tikzbracemark{ml-end}$
% \end{algorithm}
%
% \AddNote{prox-start}{prox-end}{prox-start}{\small Proximal\\Decoding}
% \AddNote{ml-start}{ml-end}{ml-start}{\small ML-on-List}
% \end{minipage}%
% \begin{minipage}{.6\textwidth}
% \centering
% \begin{figure}[H]
% \centering
% \vspace*{-12mm}
%
% \begin{tikzpicture}[scale=0.42]
% \begin{axis}[
% grid=both,
% xlabel={SNR}, ylabel={BER},
% ymode=log,
% legend style={at={(0.05,0.05)},anchor=south west},
% ymax=1.5, ymin=3e-8,
% ]
%
% \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
% {res/2d_ber_fer_dfr_20433484_proximal.csv};
% \addlegendentry{proximal}
% \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
% {res/2d_ber_fer_dfr_20433484_hybrid.csv};
% \addlegendentry{hybrid prox. \& ML}
% \end{axis}
% \end{tikzpicture}
% \begin{tikzpicture}[scale=0.42]
% \begin{axis}[
% grid=both,
% xlabel={SNR}, ylabel={FER},
% ymode=log,
% legend style={at={(0.05,0.05)},anchor=south west},
% ymax=1.5, ymin=3e-8,
% ]
%
% \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
% {res/2d_ber_fer_dfr_20433484_proximal.csv};
% \addlegendentry{proximal}
% \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
% {res/2d_ber_fer_dfr_20433484_hybrid.csv};
% \addlegendentry{hybrid prox. \& ML}
% \end{axis}
% \end{tikzpicture}\\
% \begin{tikzpicture}[scale=0.42]
% \begin{axis}[
% grid=both,
% xlabel={SNR}, ylabel={Decoding Failure Rate},
% ymode=log,
% legend style={at={(0.05,0.05)},anchor=south west},
% ymax=1.5, ymin=3e-8,
% ]
%
% \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
% {res/2d_ber_fer_dfr_20433484_proximal.csv};
% \addlegendentry{proximal}
% \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
% {res/2d_ber_fer_dfr_20433484_hybrid.csv};
% \addlegendentry{hybrid prox. \& ML}
% \end{axis}
% \end{tikzpicture}
%
% \caption{Simulation results for $\gamma = 0.05, \omega = 0.05, K=200, N=12$}
% \label{fig:simulation_results_hybrid}
% \end{figure}
% \end{minipage}
%\end{frame}
\newcommand*{\AddNote}[4]{%
\begin{tikzpicture}[overlay, remember picture]
\draw [decoration={brace,amplitude=0.5em},decorate,ultra thick]
($(#3)!([yshift=1.5ex]#1)!($(#3)-(0,1)$)$) --
($(#3)!(#2)!($(#3)-(0,1)$)$)
node [align=center, text width=2cm, pos=0.5, anchor=west] {#4};
\end{tikzpicture}
}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t, fragile]
@ -252,96 +583,56 @@ Output $\boldsymbol{\hat{x}}$
\setcounter{footnote}{0}
\begin{itemize}
\item Comparison of proximal \& hybrid-proximal-ML\\
decoding simulation
\footnote{(3,6) regular LDPC Code with $n=204, k=102$
\cite[Code: 204.33.484]{mackay_enc}}
results
\item Improvement of proximal decoding by adding an ``ML-on-list'' step after iterating
\end{itemize}
\begin{minipage}{.4\textwidth}
\begin{minipage}[t]{.48\textwidth}
\centering
\begin{algorithm}[caption={}, label={},
basicstyle=\fontsize{6.5}{7.5}\selectfont
\begin{figure}
\centering
\begin{algorithm}[caption={}, label={},
basicstyle=\fontsize{7.5}{9.5}\selectfont
]
$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$$\hspace{4.185cm}\tikzbracemark{prox-start}$
$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$
for $k=0$ to $K-1$ do
$\boldsymbol{r}^{\left( k+1 \right)} = \boldsymbol{s}^{(k)} - \omega \nabla L \left( \boldsymbol{s}^{(k)}; \boldsymbol{y} \right) $
Compute $\nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)$
$\boldsymbol{s}^{\left( k+1 \right)} = \boldsymbol{r}^{(k+1)} - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right) $
$\boldsymbol{\hat{x}} = \text{sign}\left( \boldsymbol{s}^{\left( k+1 \right) } \right) $
If $\boldsymbol{\hat{x}}$ passes the parity check condition, output $\boldsymbol{\hat{x}}$
end for $\tikzbracemark{prox-end}$
Find $N$ most probably wrong bits $\hspace{2cm}\tikzbracemark{ml-start}$
Generate variations $\boldsymbol{\tilde{x}}_n$ of $\boldsymbol{\hat{x}}$ with the $N$ bits modified
Compute $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right) \forall n \in \left[ 1 : N-1 \right] $
Output $\boldsymbol{\tilde{x}}_n$ with lowest $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)$ $\tikzbracemark{ml-end}$
\end{algorithm}
If $\boldsymbol{\hat{x}}$ passes the parity check condition, break the loop.
end for
Output $\boldsymbol{\hat{x}}$
\end{algorithm}
\caption{Proximal decoding algorithm \cite{proximal_paper}}
\end{figure}
\AddNote{prox-start}{prox-end}{prox-start}{\small Proximal\\Decoding}
\AddNote{ml-start}{ml-end}{ml-start}{\small ML-on-List}
\end{minipage}%
\begin{minipage}{.6\textwidth}
\hfill\begin{minipage}[t]{.48\textwidth}
\centering
\begin{figure}[H]
\centering
\vspace*{-12mm}
\begin{tikzpicture}[scale=0.42]
\begin{axis}[
grid=both,
xlabel={SNR}, ylabel={BER},
ymode=log,
legend style={at={(0.05,0.05)},anchor=south west},
ymax=1.5, ymin=3e-8,
]
\begin{figure}
\centering
\addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
{res/2d_ber_fer_dfr_20433484_proximal.csv};
\addlegendentry{proximal}
\addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
{res/2d_ber_fer_dfr_20433484_hybrid.csv};
\addlegendentry{hybrid prox. \& ML}
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}[scale=0.42]
\begin{axis}[
grid=both,
xlabel={SNR}, ylabel={FER},
ymode=log,
legend style={at={(0.05,0.05)},anchor=south west},
ymax=1.5, ymin=3e-8,
]
\begin{algorithm}[caption={}, label={},
basicstyle=\fontsize{7.5}{9.5}\selectfont
]
$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$
for $k=0$ to $K-1$ do
$\boldsymbol{r}^{\left( k+1 \right)} = \boldsymbol{s}^{(k)} - \omega \nabla L \left( \boldsymbol{s}^{(k)}; \boldsymbol{y} \right) $
Compute $\nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)$
$\boldsymbol{s}^{\left( k+1 \right)} = \boldsymbol{r}^{(k+1)} - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right) $
$\boldsymbol{\hat{x}} = \text{sign}\left( \boldsymbol{s}^{\left( k+1 \right) } \right) $
$\textcolor{KITblue}{\text{If }\boldsymbol{\hat{x}}\text{ passes the parity check condition, output }\boldsymbol{\hat{x}}}$
end for
$\textcolor{KITblue}{\text{Find }N\text{ most probably wrong bits.}}$
$\textcolor{KITblue}{\text{Generate variations } \boldsymbol{\tilde{x}}_n\text{ of }\boldsymbol{\hat{x}}\text{ with the }N\text{ bits modified.}}$
$\textcolor{KITblue}{\text{Compute }d_H\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right) \forall n \in \left[ 1 : N-1 \right]}$
$\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{x}}_n\text{ with lowest }d_H\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)}$
\end{algorithm}
\addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
{res/2d_ber_fer_dfr_20433484_proximal.csv};
\addlegendentry{proximal}
\addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
{res/2d_ber_fer_dfr_20433484_hybrid.csv};
\addlegendentry{hybrid prox. \& ML}
\end{axis}
\end{tikzpicture}\\
\begin{tikzpicture}[scale=0.42]
\begin{axis}[
grid=both,
xlabel={SNR}, ylabel={Decoding Failure Rate},
ymode=log,
legend style={at={(0.05,0.05)},anchor=south west},
ymax=1.5, ymin=3e-8,
]
\addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
{res/2d_ber_fer_dfr_20433484_proximal.csv};
\addlegendentry{proximal}
\addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
{res/2d_ber_fer_dfr_20433484_hybrid.csv};
\addlegendentry{hybrid prox. \& ML}
\end{axis}
\end{tikzpicture}
\caption{Simulation results for $\gamma = 0.05, \omega = 0.05, K=200, N=12$}
\label{fig:simulation_results_hybrid}
\caption{Hybrid proximal \& ML decoding algorithm}
\end{figure}
\end{minipage}
\end{frame}
@ -376,7 +667,7 @@ end for
Output $\boldsymbol{\hat{x}}$
\end{algorithm}
\caption{Proximal decoding algorithm}
\caption{Proximal decoding algorithm \cite{proximal_paper}}
\end{figure}
\end{minipage}%
@ -398,8 +689,8 @@ for $k=0$ to $K-1$ do
end for
$\textcolor{KITblue}{\text{Find }N\text{ most probably wrong bits.}}$
$\textcolor{KITblue}{\text{Generate variations } \boldsymbol{\tilde{x}}_n\text{ of }\boldsymbol{\hat{x}}\text{ with the }N\text{ bits modified.}}$
$\textcolor{KITblue}{\text{Compute }d_H\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right) \forall n \in \left[ 1 : N-1 \right]}$
$\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{x}}_n\text{ with lowest }d_H\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)}$
$\textcolor{KITblue}{\text{Compute }\langle \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \rangle \forall n \in \left[ 1 : N-1 \right]}$
$\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{x}}_n\text{ with lowest }\langle \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \rangle}$
\end{algorithm}
\caption{Hybrid proximal \& ML decoding algorithm}