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Added heatmaps showing relationship between gamma and omega

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Andreas Tsouchlos
2023-04-11 15:24:09 +02:00
parent c80d37e6b3
commit f884042269
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latex/thesis/chapters/appendix.tex
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@@ -1,11 +1,11 @@
\appendix
\chapter{A Comparison of the Behaviour of Various Codes}
\chapter{A Comparison of the Behaviour of Proximal Decoding for Various Codes}
\begin{figure}[H]
\centering
\begin{subfigure}[c]{0.48\textwidth}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[view={75}{30},
@@ -38,7 +38,7 @@
\cite[\text{96.3.965}]{mackay_enc}}
\end{subfigure}%
\hfill
\begin{subfigure}[c]{0.48\textwidth}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[view={75}{30},
@@ -70,7 +70,9 @@
\caption{BCH code with $n=31, k=26$\\[2\baselineskip]}
\end{subfigure}
\begin{subfigure}[c]{0.48\textwidth}
\vspace{3mm}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[view={75}{30},
@@ -81,29 +83,29 @@
width=\textwidth,
height=0.75\textwidth,]
\addplot3[surf,
mesh/rows=17, mesh/cols=14,
mesh/rows=17, mesh/cols=10,
colormap/viridis] table [col sep=comma,
x=SNR, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_20433484.csv};
{res/proximal/2d_ber_fer_dfr_20433484_fewer_SNR.csv};
\addplot3[RedOrange, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.05},
x=SNR, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_20433484.csv};
{res/proximal/2d_ber_fer_dfr_20433484_fewer_SNR.csv};
\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.01},
x=SNR, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_20433484.csv};
{res/proximal/2d_ber_fer_dfr_20433484_fewer_SNR.csv};
\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
discard if not={gamma}{0.15},
x=SNR, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_20433484.csv};
{res/proximal/2d_ber_fer_dfr_20433484_fewer_SNR.csv};
\end{axis}
\end{tikzpicture}
\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc}}
\end{subfigure}%
\hfill
\begin{subfigure}[c]{0.48\textwidth}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[view={75}{30},
@@ -136,7 +138,9 @@
\cite[\text{204.55.187}]{mackay_enc}}
\end{subfigure}%
\begin{subfigure}[c]{0.48\textwidth}
\vspace{3mm}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[view={75}{30},
@@ -169,7 +173,7 @@
\cite[\text{408.33.844}]{mackay_enc}}
\end{subfigure}%
\hfill
\begin{subfigure}[c]{0.48\textwidth}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[view={75}{30},
@@ -202,9 +206,9 @@
\cite[\text{PEGReg252x504}]{mackay_enc}}
\end{subfigure}%
\vspace{1cm}
\vspace{5mm}
\begin{subfigure}[c]{\textwidth}
\begin{subfigure}[t]{\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[hide axis,
@@ -231,7 +235,7 @@
\begin{figure}[H]
\centering
\begin{subfigure}[c]{0.48\textwidth}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
@@ -273,7 +277,7 @@
\cite[\text{96.3.965}]{mackay_enc}}
\end{subfigure}%
\hfill%
\begin{subfigure}[c]{0.48\textwidth}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
@@ -315,7 +319,9 @@
\caption{BCH code with $n=31, k=26$\\[\baselineskip]}
\end{subfigure}%
\begin{subfigure}[c]{0.48\textwidth}
\vspace{3mm}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
@@ -369,7 +375,7 @@
\cite[\text{204.33.484}]{mackay_enc}}
\end{subfigure}%
\hfill%
\begin{subfigure}[c]{0.48\textwidth}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
@@ -412,7 +418,9 @@
\cite[\text{204.55.187}]{mackay_enc}}
\end{subfigure}%
\begin{subfigure}[c]{0.48\textwidth}
\vspace{3mm}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
@@ -455,7 +463,7 @@
\cite[\text{408.33.844}]{mackay_enc}}
\end{subfigure}%
\hfill%
\begin{subfigure}[c]{0.48\textwidth}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
@@ -497,9 +505,9 @@
\label{fig:prox:improved:comp:504}
\end{subfigure}%
\vspace{1cm}
\vspace{5mm}
\begin{subfigure}[c]{\textwidth}
\begin{subfigure}[t]{\textwidth}
\centering
\begin{tikzpicture}
@@ -534,172 +542,401 @@
\label{fig:prox:improved:comp}
\end{figure}
\chapter{\acs{LP} Decoding using \acs{ADMM} as a Proximal Algorithm}%
\label{chapter:LD Decoding using ADMM as a Proximal Algorithm}
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.48\textwidth}
\centering
\todo{Find out how to properly title and section appendix}
\begin{tikzpicture}
\begin{axis}[
colormap/viridis,
xlabel={$\omega$}, ylabel={$\gamma$},
at={(0,0)}, view={0}{90},
zmode=log,
ytick={0, 0.05, 0.1, 0.15},
yticklabels={0, 0.05, 0.1, 0.15},
xtick={0.05, 0.1, 0.15, 0.2},
xticklabels={0.05, 0.1, 0.15, 0.2},
width=\textwidth,
height=0.75\textwidth,
point meta min=-5.7,
point meta max=-0.5,
]
\addplot3[
surf,
shader=flat,
mesh/rows=17, mesh/cols=10,
]
table [col sep=comma, x=omega, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_gamma_omega_963965.csv};
\end{axis}
\end{tikzpicture}
\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=96, k=48$
\cite[\text{96.3.965}]{mackay_enc}}
\end{subfigure}%
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\centering
%For problems of the form%
\begin{tikzpicture}
\begin{axis}[
colormap/viridis,
xlabel={$\omega$}, ylabel={$\gamma$},
at={(0,0)}, view={0}{90},
zmode=log,
ytick={0, 0.05, 0.1, 0.15},
yticklabels={0, 0.05, 0.1, 0.15},
xtick={0.05, 0.1, 0.15, 0.2},
xticklabels={0.05, 0.1, 0.15, 0.2},
width=\textwidth,
height=0.75\textwidth,
point meta min=-5.7,
point meta max=-0.5,
]
\addplot3[
surf,
shader=flat,
mesh/rows=17, mesh/cols=10,
]
table [col sep=comma, x=omega, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_gamma_omega_bch_31_26.csv};
\end{axis}
\end{tikzpicture}
\caption{BCH code with $n=31, k=26$\\[2\baselineskip]}
\end{subfigure}%
\vspace{3mm}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
colormap/viridis,
xlabel={$\omega$}, ylabel={$\gamma$},
at={(0,0)}, view={0}{90},
zmode=log,
ytick={0, 0.05, 0.1, 0.15},
yticklabels={0, 0.05, 0.1, 0.15},
xtick={0.05, 0.1, 0.15, 0.2},
xticklabels={0.05, 0.1, 0.15, 0.2},
width=\textwidth,
height=0.75\textwidth,
point meta min=-5.7,
point meta max=-0.5,
]
\addplot3[
surf,
shader=flat,
mesh/rows=17, mesh/cols=10,
]
table [col sep=comma, x=omega, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_gamma_omega_20433484.csv};
\end{axis}
\end{tikzpicture}
\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc}}
\end{subfigure}%
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
colormap/viridis,
xlabel={$\omega$}, ylabel={$\gamma$},
at={(0,0)}, view={0}{90},
zmode=log,
ytick={0, 0.05, 0.1, 0.15},
yticklabels={0, 0.05, 0.1, 0.15},
xtick={0.05, 0.1, 0.15, 0.2},
xticklabels={0.05, 0.1, 0.15, 0.2},
width=\textwidth,
height=0.75\textwidth,
point meta min=-5.7,
point meta max=-0.5,
]
\addplot3[
surf,
shader=flat,
mesh/rows=17, mesh/cols=10,
]
table [col sep=comma, x=omega, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_gamma_omega_20455187.csv};
\end{axis}
\end{tikzpicture}
\caption{$\left( 5, 10 \right)$-regular \ac{LDPC} code with $n=204, k=102$
\cite[\text{204.55.187}]{mackay_enc}}
\end{subfigure}%
\vspace{3mm}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
colormap/viridis,
xlabel={$\omega$}, ylabel={$\gamma$},
at={(0,0)}, view={0}{90},
zmode=log,
ytick={0, 0.05, 0.1, 0.15},
yticklabels={0, 0.05, 0.1, 0.15},
xtick={0.05, 0.1, 0.15, 0.2},
xticklabels={0.05, 0.1, 0.15, 0.2},
width=\textwidth,
height=0.75\textwidth,
point meta min=-5.7,
point meta max=-0.5,
]
\addplot3[
surf,
shader=flat,
mesh/rows=17, mesh/cols=10,
]
table [col sep=comma, x=omega, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_gamma_omega_40833844.csv};
\end{axis}
\end{tikzpicture}
\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=408, k=204$
\cite[\text{408.33.844}]{mackay_enc}}
\end{subfigure}%
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
colormap/viridis,
xlabel={$\omega$}, ylabel={$\gamma$},
at={(0,0)}, view={0}{90},
zmode=log,
ytick={0, 0.05, 0.1, 0.15},
yticklabels={0, 0.05, 0.1, 0.15},
xtick={0.05, 0.1, 0.15, 0.2},
xticklabels={0.05, 0.1, 0.15, 0.2},
width=\textwidth,
height=0.75\textwidth,
point meta min=-5.7,
point meta max=-0.5,
]
\addplot3[
surf,
shader=flat,
mesh/rows=17, mesh/cols=10,
]
table [col sep=comma, x=omega, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_gamma_omega_pegreg252x504.csv};
\end{axis}
\end{tikzpicture}
\caption{LDPC code (Progressive Edge Growth Construction) with $n=504, k=252$
\cite[\text{PEGReg252x504}]{mackay_enc}}
\end{subfigure}%
\vspace{5mm}
\begin{subfigure}{\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
hide axis,
scale only axis,
height=0pt,
width=0pt,
colormap/viridis,
colorbar horizontal,
point meta min=-5.7,
point meta max=-0.5,
colorbar style={
title={BER},
width=10cm,
xtick={-5,-4,...,-1},
xticklabels={$10^{-5}$,$10^{-4}$,$10^{-3}$,$10^{-2}$,$10^{-1}$}
}]
\addplot [draw=none] coordinates {(0,0)};
\end{axis}
\end{tikzpicture}
\end{subfigure}%
\caption{}
\label{fig:prox:gamma_omega_multiple}
\end{figure}
%\chapter{\acs{LP} Decoding using \acs{ADMM} as a Proximal Algorithm}%
%\label{chapter:LD Decoding using ADMM as a Proximal Algorithm}
%
%\todo{Find out how to properly title and section appendix}
%
%%For problems of the form%
%%\begin{align*}
%% \text{minimize}\hspace{2mm} & f\left( \boldsymbol{x} \right)
%% + g\left( \boldsymbol{A}\boldsymbol{x} \right) \\
%% \text{subject to}\hspace{2mm} & \boldsymbol{x} \in \mathbb{R}^n
%%,\end{align*}%
%%%
%%a version of \ac{ADMM}, \textit{linearized \ac{ADMM}}, can be expressed
%%as a proximal algorithm \cite[Sec. 4.4.2]{proximal_algorithms}:
%%%
%%\begin{align*}
%% \boldsymbol{x} &\leftarrow \textbf{prox}_{\mu f}\left( \boldsymbol{x}
%% - \frac{\mu}{\lambda}\boldsymbol{A}^\text{T}\left( \boldsymbol{A}\boldsymbol{x}
%% - \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
%% \boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left( \boldsymbol{A}\boldsymbol{x}
%% + \boldsymbol{u} \right) \\
%% \boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{A} \boldsymbol{x} - \boldsymbol{z}
%%.\end{align*}
%
%In order to express \ac{LP} decoding using \ac{ADMM} through proximal operators,
%it can be rewritten to fit the template for \textit{linearized \ac{ADMM}} given
%in \cite[Sec. 4.4.2]{proximal_algorithms}.
%We start with the general formulation of the \ac{LP} decoding problem:%
%%
%\begin{align*}
% \text{minimize}\hspace{2mm} & f\left( \boldsymbol{x} \right)
% + g\left( \boldsymbol{A}\boldsymbol{x} \right) \\
% \text{subject to}\hspace{2mm} & \boldsymbol{x} \in \mathbb{R}^n
% \text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}} \\
% \text{subject to}\hspace{2mm} & \boldsymbol{T}_j \tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}
% \hspace{5mm} \forall j \in \mathcal{J}
%.\end{align*}
%%
%The constraints can be moved into the objective function: %
%%
%\begin{align}
% \begin{aligned}
% \text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
% + \sum_{j\in \mathcal{J}} I_{P_{d_j}}\left(
% \boldsymbol{T}_j\tilde{\boldsymbol{c}} \right) \\
% \text{subject to}\hspace{2mm} & \tilde{\boldsymbol{c}} \in \mathbb{R}^n,
% \end{aligned}
% \label{eq:app:sum_reformulated}
%\end{align}%
%%
%using the \textit{indicator functions}
%$I_{\mathcal{P}_{d_j}} : \mathbb{R}^{d_j} \rightarrow \left\{ 0, +\infty \right\},
%\hspace{3mm} j\in \mathcal{J}$, defined as%
%%
%\begin{align*}
% I_{\mathcal{P}_{d_j}}\left( \boldsymbol{t} \right) :=
% \begin{cases}
% 0 & \boldsymbol{t} \in \mathcal{P}_{d_j} \\
% +\infty & \boldsymbol{t} \not\in \mathcal{P}_{d_j}
% \end{cases}
%.\end{align*}%
%%
%Further defining
%%
%\begin{align*}
% \boldsymbol{T} := \begin{bmatrix}
% \boldsymbol{T}_1 \\
% \boldsymbol{T}_2 \\
% \vdots \\
% \boldsymbol{T}_m
% \end{bmatrix}
% \hspace{5mm}\text{and}\hspace{5mm}
% g\left( \boldsymbol{t} \right) = \sum_{j\in\mathcal{J}} I_{\mathcal{P}_{d_j}}\left(
% \boldsymbol{B}_j \boldsymbol{t} \right)
%,\end{align*}%
%%
%a version of \ac{ADMM}, \textit{linearized \ac{ADMM}}, can be expressed
%as a proximal algorithm \cite[Sec. 4.4.2]{proximal_algorithms}:
%\todo{Define $\boldsymbol{B}_j$}%
%problem (\ref{eq:app:sum_reformulated}) becomes%
%%
%\begin{align}
% \begin{aligned}
% \text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
% + g\left( \boldsymbol{T}\tilde{\boldsymbol{c}} \right) \\
% \text{subject to}\hspace{2mm} & \tilde{\boldsymbol{c}} \in \mathbb{R}^n.
% \end{aligned}
% \label{eq:app:func_reformulated}
%\end{align}
%%
%\todo{Fix $\mu f$ and $\lambda g$ in the steps below}%
%In this form, it fits the template for linearized \ac{ADMM}.
%The iterative algorithm can then be expressed as%
%%
%\begin{align}
% \begin{aligned}
% \tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
% - \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
% - \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
% \boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left(\boldsymbol{T}\tilde{\boldsymbol{c}}
% + \boldsymbol{u} \right) \\
% \boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{T} \tilde{\boldsymbol{c}}
% - \boldsymbol{z}.
% \end{aligned}
% \label{eq:app:admm_prox}
%\end{align}
%%
%
%Using the definition of the proximal operator, the $\tilde{\boldsymbol{c}}$ update step
%can be rewritten to match the definition given in section \ref{sec:lp:Decoding Algorithm}:%
%%
%\begin{align*}
% \boldsymbol{x} &\leftarrow \textbf{prox}_{\mu f}\left( \boldsymbol{x}
% - \frac{\mu}{\lambda}\boldsymbol{A}^\text{T}\left( \boldsymbol{A}\boldsymbol{x}
% \tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
% - \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
% - \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
% \boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left( \boldsymbol{A}\boldsymbol{x}
% + \boldsymbol{u} \right) \\
% \boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{A} \boldsymbol{x} - \boldsymbol{z}
% &= \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
% - \frac{\mu}{2} \left\Vert \boldsymbol{T}^\text{T}
% \left( \boldsymbol{T}\tilde{\boldsymbol{c}}
% - \boldsymbol{z} + \boldsymbol{u} \right) \right\Vert_2^2 \right) \\
% &\overset{\text{(a)}}{=} \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}
% \tilde{\boldsymbol{c}}
% - \frac{\mu}{2} \left\Vert \boldsymbol{T}\tilde{\boldsymbol{c}}
% - \boldsymbol{z} + \boldsymbol{u} \right\Vert_2^2 \right) \\
% &= \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
% - \frac{\mu}{2} \left\Vert \begin{bmatrix}
% \boldsymbol{T}_1 \\
% \boldsymbol{T}_2 \\
% \vdots \\
% \boldsymbol{T}_m
% \end{bmatrix}
% \tilde{\boldsymbol{c}}
% - \begin{bmatrix}
% \boldsymbol{z}_1 \\
% \boldsymbol{z}_2 \\
% \vdots \\
% \boldsymbol{z}_m
% \end{bmatrix}
% + \begin{bmatrix}
% \boldsymbol{u}_1 \\
% \boldsymbol{u}_2 \\
% \vdots \\
% \boldsymbol{u}_m
% \end{bmatrix} \right\Vert_2^2 \right),
% \hspace{5mm}\boldsymbol{z}_j,\boldsymbol{u}_j \in \mathbb{F}_2^{d_j},
% \hspace{2mm} j\in\mathcal{J}\\
% &= \argmin_{\tilde{\boldsymbol{c}}} \left( \boldsymbol{\gamma}^\text{T} \tilde{\boldsymbol{c}}
% - \frac{\mu}{2} \sum_{j \in J} \left\Vert \boldsymbol{T}_j \tilde{\boldsymbol{c}}
% - \boldsymbol{z}_j + \boldsymbol{u}_j \right\Vert_2^2 \right)
%.\end{align*}
In order to express \ac{LP} decoding using \ac{ADMM} through proximal operators,
it can be rewritten to fit the template for \textit{linearized \ac{ADMM}} given
in \cite[Sec. 4.4.2]{proximal_algorithms}.
We start with the general formulation of the \ac{LP} decoding problem:%
%
\begin{align*}
\text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}} \\
\text{subject to}\hspace{2mm} & \boldsymbol{T}_j \tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}
\hspace{5mm} \forall j \in \mathcal{J}
.\end{align*}
%
The constraints can be moved into the objective function: %
%
\begin{align}
\begin{aligned}
\text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
+ \sum_{j\in \mathcal{J}} I_{P_{d_j}}\left(
\boldsymbol{T}_j\tilde{\boldsymbol{c}} \right) \\
\text{subject to}\hspace{2mm} & \tilde{\boldsymbol{c}} \in \mathbb{R}^n,
\end{aligned}
\label{eq:app:sum_reformulated}
\end{align}%
%
using the \textit{indicator functions}
$I_{\mathcal{P}_{d_j}} : \mathbb{R}^{d_j} \rightarrow \left\{ 0, +\infty \right\},
\hspace{3mm} j\in \mathcal{J}$, defined as%
%
\begin{align*}
I_{\mathcal{P}_{d_j}}\left( \boldsymbol{t} \right) :=
\begin{cases}
0 & \boldsymbol{t} \in \mathcal{P}_{d_j} \\
+\infty & \boldsymbol{t} \not\in \mathcal{P}_{d_j}
\end{cases}
.\end{align*}%
%
Further defining
%
\begin{align*}
\boldsymbol{T} := \begin{bmatrix}
\boldsymbol{T}_1 \\
\boldsymbol{T}_2 \\
\vdots \\
\boldsymbol{T}_m
\end{bmatrix}
\hspace{5mm}\text{and}\hspace{5mm}
g\left( \boldsymbol{t} \right) = \sum_{j\in\mathcal{J}} I_{\mathcal{P}_{d_j}}\left(
\boldsymbol{B}_j \boldsymbol{t} \right)
,\end{align*}%
%
\todo{Define $\boldsymbol{B}_j$}%
problem (\ref{eq:app:sum_reformulated}) becomes%
%
\begin{align}
\begin{aligned}
\text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
+ g\left( \boldsymbol{T}\tilde{\boldsymbol{c}} \right) \\
\text{subject to}\hspace{2mm} & \tilde{\boldsymbol{c}} \in \mathbb{R}^n.
\end{aligned}
\label{eq:app:func_reformulated}
\end{align}
%
\todo{Fix $\mu f$ and $\lambda g$ in the steps below}%
In this form, it fits the template for linearized \ac{ADMM}.
The iterative algorithm can then be expressed as%
%
\begin{align}
\begin{aligned}
\tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
- \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
- \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
\boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left(\boldsymbol{T}\tilde{\boldsymbol{c}}
+ \boldsymbol{u} \right) \\
\boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{T} \tilde{\boldsymbol{c}}
- \boldsymbol{z}.
\end{aligned}
\label{eq:app:admm_prox}
\end{align}
%
Using the definition of the proximal operator, the $\tilde{\boldsymbol{c}}$ update step
can be rewritten to match the definition given in section \ref{sec:lp:Decoding Algorithm}:%
%
\begin{align*}
\tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
- \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
- \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
&= \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
- \frac{\mu}{2} \left\Vert \boldsymbol{T}^\text{T}
\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
- \boldsymbol{z} + \boldsymbol{u} \right) \right\Vert_2^2 \right) \\
&\overset{\text{(a)}}{=} \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}
\tilde{\boldsymbol{c}}
- \frac{\mu}{2} \left\Vert \boldsymbol{T}\tilde{\boldsymbol{c}}
- \boldsymbol{z} + \boldsymbol{u} \right\Vert_2^2 \right) \\
&= \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
- \frac{\mu}{2} \left\Vert \begin{bmatrix}
\boldsymbol{T}_1 \\
\boldsymbol{T}_2 \\
\vdots \\
\boldsymbol{T}_m
\end{bmatrix}
\tilde{\boldsymbol{c}}
- \begin{bmatrix}
\boldsymbol{z}_1 \\
\boldsymbol{z}_2 \\
\vdots \\
\boldsymbol{z}_m
\end{bmatrix}
+ \begin{bmatrix}
\boldsymbol{u}_1 \\
\boldsymbol{u}_2 \\
\vdots \\
\boldsymbol{u}_m
\end{bmatrix} \right\Vert_2^2 \right),
\hspace{5mm}\boldsymbol{z}_j,\boldsymbol{u}_j \in \mathbb{F}_2^{d_j},
\hspace{2mm} j\in\mathcal{J}\\
&= \argmin_{\tilde{\boldsymbol{c}}} \left( \boldsymbol{\gamma}^\text{T} \tilde{\boldsymbol{c}}
- \frac{\mu}{2} \sum_{j \in J} \left\Vert \boldsymbol{T}_j \tilde{\boldsymbol{c}}
- \boldsymbol{z}_j + \boldsymbol{u}_j \right\Vert_2^2 \right)
.\end{align*}
%
Step (a) can be justified by observing that multiplication with $\boldsymbol{T}^\text{T}$
only reorders components, leaving their values unchanged.
Similarly to the $\boldsymbol{c}$ update, the $\boldsymbol{z}$ update step can be rewritten.
Since $g\left( \cdot \right)$ is separable, so is its proximal operator
\cite[Sec. 2.1]{proximal_algorithms}. The $\boldsymbol{z}$ update step can then
be expressed as a number of smaller steps:%
%
\begin{gather*}
\boldsymbol{z} \leftarrow \textbf{prox}_{\lambda g} \left(\boldsymbol{T}\tilde{\boldsymbol{c}}
+ \boldsymbol{u} \right) \\[0.5em]
\iff \\[0.5em]
\begin{alignedat}{3}
\boldsymbol{z}_j &\leftarrow \textbf{prox}_{\lambda I_{\mathcal{P}_{d_j}}}\left(
\boldsymbol{T}_j \tilde{\boldsymbol{c}} + \boldsymbol{u}_j \right),
\hspace{5mm} && \forall j\in\mathcal{J} \\
& \overset{\text{(b)}}{=} \Pi_{\mathcal{P}_{d_j}}\left( \boldsymbol{T}_j
\tilde{\boldsymbol{c}}
+ \boldsymbol{u}_j \right), \hspace{5mm} && \forall j\in\mathcal{J}
,\end{alignedat}
\end{gather*}
%
where (b) results from the fact that appying the proximal operator on the
indicator function of a convex set amounts to a projection onto the set
\cite[Sec. 1.2]{proximal_algorithms}.
%%
%Step (a) can be justified by observing that multiplication with $\boldsymbol{T}^\text{T}$
%only reorders components, leaving their values unchanged.
%Similarly to the $\boldsymbol{c}$ update, the $\boldsymbol{z}$ update step can be rewritten.
%Since $g\left( \cdot \right)$ is separable, so is its proximal operator
%\cite[Sec. 2.1]{proximal_algorithms}. The $\boldsymbol{z}$ update step can then
%be expressed as a number of smaller steps:%
%%
%\begin{gather*}
% \boldsymbol{z} \leftarrow \textbf{prox}_{\lambda g} \left(\boldsymbol{T}\tilde{\boldsymbol{c}}
% + \boldsymbol{u} \right) \\[0.5em]
% \iff \\[0.5em]
% \begin{alignedat}{3}
% \boldsymbol{z}_j &\leftarrow \textbf{prox}_{\lambda I_{\mathcal{P}_{d_j}}}\left(
% \boldsymbol{T}_j \tilde{\boldsymbol{c}} + \boldsymbol{u}_j \right),
% \hspace{5mm} && \forall j\in\mathcal{J} \\
% & \overset{\text{(b)}}{=} \Pi_{\mathcal{P}_{d_j}}\left( \boldsymbol{T}_j
% \tilde{\boldsymbol{c}}
% + \boldsymbol{u}_j \right), \hspace{5mm} && \forall j\in\mathcal{J}
% ,\end{alignedat}
%\end{gather*}
%%
%where (b) results from the fact that appying the proximal operator on the
%indicator function of a convex set amounts to a projection onto the set
%\cite[Sec. 1.2]{proximal_algorithms}.

54
latex/thesis/chapters/proximal_decoding.tex
View File

@@ -260,7 +260,7 @@ It was subsequently reimplemented in C++ using the Eigen%
linear algebra library to achieve higher performance.
The focus has been set on a fast implementation, sometimes at the expense of
memory usage, somewhat limiting the size of the codes the implemenation can be
used with \todo{Is this a appropriate for a bachelor's thesis?}.
used with \todo{Is this sentence appropriate for a bachelor's thesis?}.
The evaluation of the simulation results has been wholly realized in Python.
The gradient of the code-constraint polynomial \cite[Sec. 2.3]{proximal_paper}
@@ -498,16 +498,60 @@ The parameter $\gamma$ describes the step-size for the optimization step
dealing with the code-constraint polynomial;
the parameter $\omega$ describes the step-size for the step dealing with the
negative-log likelihood.
The relationship between $\omega$ and $\gamma$ is studied in figure
\ref{TODO}.
The relationship between $\omega$ and $\gamma$ is portrayed in figure
\ref{fig:prox:gamma_omega}.
The \ac{SNR} is kept constant at $\SI{4}{dB}$.
Similar behaviour to $\gamma$ is exhibited: the \ac{BER} is minimized when
keeping the value within certain bounds, without displaying a clear
optimum.
It is noteworthy that the decoder seems to achieve the best performance for
similar values of the two step sizes.
Again, this consideration applies to a multitude of different codes, depicted
in figure \ref{TODO}.
Again, this consideration applies to a multitude of different codes, as
depicted in figure \ref{fig:prox:gamma_omega_multiple}.
\begin{figure}[H]
\centering
\begin{tikzpicture}
\begin{axis}[
colormap/viridis,
colorbar,
xlabel={$\omega$}, ylabel={$\gamma$},
at={(0,0)}, view={0}{90},
zmode=log,
ytick={0, 0.05, 0.1, 0.15},
yticklabels={0, 0.05, 0.1, 0.15},
xtick={0.05, 0.1, 0.15, 0.2},
xticklabels={0.05, 0.1, 0.15, 0.2},
width=0.6\textwidth,
height=0.45\textwidth,
point meta min=-5.7,
point meta max=-0.5,
colorbar style={
title={BER},
ytick={-5,-4,...,-1},
yticklabels={$10^{-5}$,$10^{-4}$,$10^{-3}$,$10^{-2}$,$10^{-1}$}
}]
]
\addplot3[
surf,
shader=flat,
mesh/rows=17, mesh/cols=10,
]
table [col sep=comma, x=omega, y=gamma, z=BER]
{res/proximal/2d_ber_fer_dfr_gamma_omega_20433484.csv};
\end{axis}
\end{tikzpicture}
\caption{The \ac{BER} as a function of the two step sizes\protect\footnotemark{}}
\label{fig:prox:gamma_omega}
\end{figure}%
%
\footnotetext{(3,6) regular \ac{LDPC} code with n = 204, k = 102
\cite[\text{204.33.484}]{mackay_enc}; $SNR=\SI{4}{dB}, K=100, \eta=1.5$
}%
%
To better understand how to determine the optimal value for the parameter $K$,
the average error is inspected.