Done with ADMM explanation

latex/presentations/final/presentation.bib

@@ -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}
+}
+

latex/presentations/final/sections/decoding_algorithms.tex

@@ -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}