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Done with ADMM explanation

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Andreas Tsouchlos 2023-04-12 14:00:04 +02:00
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11
latex/presentations/final/presentation.bib
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@ -121,3 +121,14 @@
doi={10.1109/TIT.2020.2984247}
}
@ARTICLE{original_admm,
author={Barman, Siddharth and Liu, Xishuo and Draper, Stark C. and Recht, Benjamin},
journal={IEEE Transactions on Information Theory},
title={Decomposition Methods for Large Scale LP Decoding},
year={2013},
volume={59},
number={12},
pages={7870-7886},
doi={10.1109/TIT.2013.2281372}
}

83
latex/presentations/final/sections/decoding_algorithms.tex
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@ -713,7 +713,7 @@ return $\boldsymbol{\hat{c}}$
\end{tikzpicture}
\begin{align*}
\text{minimize}\hspace{2mm} &\boldsymbol{\gamma}^\text{T} \boldsymbol{c} \\
\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{2mm}
\forall j\in \mathcal{J}
@ -752,6 +752,87 @@ return $\boldsymbol{\hat{c}}$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{LP Decoding using ADMM}%
\label{sub:LP Decoding using ADMM}
\begin{frame}[t]
\frametitle{LP Decoding using ADMM}
\begin{itemize}
\item Slight reformulation of the LCLP:
\begin{align*}
\begin{aligned}
\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}
\forall j\in \mathcal{J}
\end{aligned}\hspace{2mm},\hspace{1cm}
g_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*}
\item Iterative algorithm:
\begin{alignat*}{3}
\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) \\
\boldsymbol{z}_j &\leftarrow \argmin_{\boldsymbol{z}_j}
\left( g\left( \boldsymbol{z}_j \right)
+ \frac{\rho}{2} \left\Vert \boldsymbol{T}_j \tilde{\boldsymbol{c}}
- \boldsymbol{z}_j + \boldsymbol{u}_j \right\Vert \right),
\hspace{5mm} &&\forall j\in\mathcal{J} \\
\boldsymbol{u}_j &\leftarrow \boldsymbol{u}_j
+ \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j,
\hspace{5mm} &&\forall j\in\mathcal{J}
% \left( g\left( \boldsymbol{\boldsymbol{z}_j} \right)
% + \frac{\rho}{2} \left\Vert \boldsymbol{T}_j\tilde{\boldsymbol{c}}
% - \boldsymbol{z}_j + \boldsymbol{u}_j\right\Vert \right)
\end{alignat*}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{LP Decoding using ADMM}
\begin{itemize}
\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 $\left( \boldsymbol{u}_j \right)_i$}%
:
\begin{alignat*}{3}
\tilde{c}_i &\leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left(
\sum_{j\in N_v\left( i \right) } \Big( \left( \boldsymbol{z}_j \right)_i
- \left( \boldsymbol{u}_j \right)_i \Big)
- \frac{\gamma_i}{\mu} \right)
\hspace{5mm} && \forall i\in\mathcal{I} \\
\boldsymbol{z}_j &\leftarrow \Pi_{\mathcal{P}_{d_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
+ \boldsymbol{T}_j\tilde{\boldsymbol{c}}
- \boldsymbol{z}_j
\hspace{5mm} && \forall j\in\mathcal{J}
\end{alignat*}
\item The main computational effort are the projections
$\Pi_{\mathcal{P}_{d_j}}, \hspace{1mm} j\in\mathcal{J}$. Many
different approaches exist, e.g., \cite{original_admm},
\cite{efficient_lp_dec_admm}, \cite{lautern}.
\item The approach chosen here is the one described in \cite{lautern}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[t]
% \frametitle{LP Relaxation}