From bcf05d26afbf0ae1bdb4ee5f005192c2ca288461 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Sat, 15 Apr 2023 18:56:55 +0200
Subject: [PATCH] Fixed mistake in implementation details; |N_v(i)| -> d_i

---
 latex/thesis/chapters/lp_dec_using_admm.tex | 23 ++++++++++-----------
 1 file changed, 11 insertions(+), 12 deletions(-)

diff --git a/latex/thesis/chapters/lp_dec_using_admm.tex b/latex/thesis/chapters/lp_dec_using_admm.tex
index 1ad0d63..9547d6b 100644
--- a/latex/thesis/chapters/lp_dec_using_admm.tex
+++ b/latex/thesis/chapters/lp_dec_using_admm.tex
@@ -134,8 +134,8 @@ examplary code, which is described by the generator and parity-check matrices%
 and has only two possible codewords:
 %
 \begin{align*}
-\mathcal{C} = \left\{ \begin{bmatrix} 0 & 0 & 0 \end{bmatrix},
-    \begin{bmatrix} 0 & 1 & 1 \end{bmatrix}   \right\} 
+\mathcal{C} = \left\{ \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix},
+    \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}   \right\} 
 .\end{align*}
 %
 Figure \ref{fig:lp:poly:exact_ilp} shows the domain of exact \ac{ML} decoding.
@@ -631,7 +631,7 @@ The same is true for $\left( \boldsymbol{\lambda}_j \right)_i$.}
 \cite[Sec. III. B.]{original_admm}:%
 %
 \begin{alignat*}{3}
-    \tilde{c}_i &\leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( 
+    \tilde{c}_i &\leftarrow \frac{1}{d_i} \left( 
         \sum_{j\in N_v\left( i \right) } \Big( \left( \boldsymbol{z}_j \right)_i
             - \frac{1}{\mu} \left( \boldsymbol{\lambda}_j \right)_i \Big)
         - \frac{\gamma_i}{\mu} \right)
@@ -652,7 +652,7 @@ This representation can be slightly simplified by substituting
 $\boldsymbol{\lambda}_j = \mu \cdot \boldsymbol{u}_j \,\forall\,j\in\mathcal{J}$:%
 %
 \begin{alignat*}{3}
-    \tilde{c}_i &\leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( 
+    \tilde{c}_i &\leftarrow \frac{1}{d_i} \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)
@@ -692,7 +692,7 @@ while $\sum_{j\in\mathcal{J}} \lVert \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \b
             - \boldsymbol{z}_j$
     end for
     for $i$ in $\mathcal{I}$ do
-        $\tilde{c}_i \leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( 
+        $\tilde{c}_i \leftarrow \frac{1}{d_i} \left( 
             \sum_{j\in N_v\left( i \right) } \Big(
                 \left( \boldsymbol{z}_j \right)_i - \left( \boldsymbol{u}_j
                 \right)_i
@@ -724,7 +724,7 @@ The method chosen here is the one presented in \cite{lautern}.
 \section{Implementation Details}%
 \label{sec:lp:Implementation Details}
 
-The development process used to implement this decoding algorithm is the same
+The development process used to implement this decoding algorithm was the same
 as outlined in section
 \ref{sec:prox:Implementation Details} for proximal decoding.
 At first, an initial version was implemented in Python, before repeating the
@@ -752,11 +752,10 @@ Using this observation, the sum can be written as%
         - \boldsymbol{u}_j \right) \right)_i
 .\end{align*}
 Further noticing that the vectors
-$\boldsymbol{T}_j^\text{T}\left( \boldsymbol{z}_j - \boldsymbol{u}_j \right),
-\hspace{1mm} j\in\mathcal{J} $
+$\boldsymbol{T}_j^\text{T}\left( \boldsymbol{z}_j - \boldsymbol{u}_j \right)$
 unrelated to \ac{VN} $i$ have $0$ as the $i$th component, the set of indices
 the summation takes place over can be extended to $\mathcal{J}$, allowing the
-expression to be rewritten to%
+expression to be rewritten as%
 %
 \begin{align*}
     \sum_{j\in \mathcal{J}}\left( \boldsymbol{T}_j^\text{T} \left( \boldsymbol{z}_j
@@ -771,12 +770,12 @@ Defining%
     \boldsymbol{D} := \begin{bmatrix} 
         d_1 \\
         \vdots \\
-        d_m
+        d_n
     \end{bmatrix}%
     \hspace{5mm}%
     \text{and}%
     \hspace{5mm}%
-    \boldsymbol{M} := \sum_{j\in\mathcal{J}} \boldsymbol{T}_j^\text{T}
+    \boldsymbol{s} := \sum_{j\in\mathcal{J}} \boldsymbol{T}_j^\text{T}
         \left( \boldsymbol{z}_j - \boldsymbol{u}_j \right)
 \end{align*}%
 %
@@ -784,7 +783,7 @@ the $\tilde{\boldsymbol{c}}$ update can then be rewritten as%
 %
 \begin{align*}
     \tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ -1} \circ
-        \left( \boldsymbol{M} - \frac{1}{\mu}\boldsymbol{\gamma} \right)  
+        \left( \boldsymbol{s} - \frac{1}{\mu}\boldsymbol{\gamma} \right)  
 .\end{align*}
 %