From eeadb9c72661c1eaceae461c8af9580e4e5e5488 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Fri, 7 Apr 2023 16:58:22 +0200
Subject: [PATCH] Added first version of proximal implementation details

---
 latex/thesis/chapters/proximal_decoding.tex   | 61 +++++++++++++++++++
 .../chapters/theoretical_background.tex       | 11 ++++
 2 files changed, 72 insertions(+)

diff --git a/latex/thesis/chapters/proximal_decoding.tex b/latex/thesis/chapters/proximal_decoding.tex
index 00d2803..3f7099b 100644
--- a/latex/thesis/chapters/proximal_decoding.tex
+++ b/latex/thesis/chapters/proximal_decoding.tex
@@ -252,6 +252,67 @@ return $\boldsymbol{\hat{c}}$
 \section{Implementation Details}%
 \label{sec:prox:Implementation Details}
 
+The algorithm as first implemented in Python because of the fast development
+process and straightforward debugging.
+They have subsequently been reimplemented in C++ using the Eigen%
+\footnote{\url{https://eigen.tuxfamily.org}}
+linear algebra library to achieve higher performance.
+The focus has been set on a fast implementation, sometimes at the expense of
+memory usage.
+The evaluation of the simulation results has been wholly realized in Python.
+
+The gradient of the code-constraint polynomial is given by \cite[Sec. 2.3]{proximal_paper}%
+%
+\begin{align*}
+    \nabla h\left( \boldsymbol{x} \right) &= \begin{bmatrix}
+        \frac{\partial}{\partial x_1}h\left( \boldsymbol{x} \right) &
+        \ldots &
+        \frac{\partial}{\partial x_n}h\left( \boldsymbol{x} \right) &
+    \end{bmatrix}^\text{T}, \\[1em]
+    \frac{\partial}{\partial x_k}h\left( \boldsymbol{x} \right) &= 4\left( x_k^2 - 1 \right) x_k
+        + \frac{2}{x_k} \sum_{j\in N_v\left( k \right) }\left(
+            \left( \prod_{i \in N_c\left( j \right)} x_i \right)^2
+                - \prod_{i\in N_c\left( j \right) } x_i \right) 
+.\end{align*}
+%
+Evidently, the products
+$\prod_{i\in N_c\left( j \right) } x_i,\hspace{1mm}\forall i\in \mathcal{J}$
+can be precomputed, as they are the same for all components $x_k$ of $\boldsymbol{x}$.
+Defining%
+%
+\begin{align*}
+    \boldsymbol{p} := \begin{bmatrix}
+        \prod_{i\in N_c\left( 1 \right) }x_i \\
+        \vdots \\
+        \prod_{i\in N_c\left( m \right) }x_i \\
+    \end{bmatrix}
+    \hspace{5mm}
+    \text{and}
+    \hspace{5mm}
+    \boldsymbol{v} := \boldsymbol{p}^{\circ 2} - \boldsymbol{p}
+,\end{align*}
+%
+the gradient can be written as%
+%
+\begin{align*}
+    \nabla h\left( \boldsymbol{x} \right) =
+        4\left( \boldsymbol{x}^{\circ 3} - \boldsymbol{x} \right)
+        + 2\boldsymbol{x}^{\circ -1} \circ \boldsymbol{H}^\text{T}
+            \boldsymbol{v}
+,\end{align*}
+%
+enabling the computation of the gradient primarily with element-wise
+operations and matrix-vector multiplication.
+This is beneficial, as the libraries used for the implementation are
+heavily optimized for such calculations (e.g., through vectorization of the
+operations).
+\todo{Note about how the equation with which the gradient is calculated is
+itself similar to a message-passing rule}
+
+The projection $\prod_{\eta}\left( . \right)$ also proves straightforward to
+compute, as it amounts to simply clipping each component of the vector onto
+$[-\eta, \eta]$ individually.
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results}%
diff --git a/latex/thesis/chapters/theoretical_background.tex b/latex/thesis/chapters/theoretical_background.tex
index 1b21d03..ee1d77a 100644
--- a/latex/thesis/chapters/theoretical_background.tex
+++ b/latex/thesis/chapters/theoretical_background.tex
@@ -32,6 +32,17 @@ of indexed variables:%
     x_{\left[ m:n \right] } &:= \left\{ x_m, x_{m+1}, \ldots, x_{n-1}, x_n \right\} 
 .\end{align*}
 %
+In order to designate elemen-twise operations, in particular the\textit{Hadamard product}
+and the \textit{Hadamard power}, the operator $\circ$ will be used:%
+%
+\begin{alignat*}{3}
+    \boldsymbol{a} \circ \boldsymbol{b}
+        &:= \begin{bmatrix} a_1 b_1 & \ldots & a_n b_n \end{bmatrix} ^\text{T},
+    \hspace{5mm} &&\boldsymbol{a}, \boldsymbol{b} \in \mathbb{R}^n, \hspace{2mm} n\in \mathbb{N} \\
+    \boldsymbol{a}^{\circ k} &:= \begin{bmatrix} a_1^k \ldots a_n^k \end{bmatrix}^\text{T},
+    \hspace{5mm} &&\boldsymbol{a} \in \mathbb{R}^n, \hspace{2mm}k\in \mathbb{Z}
+.\end{alignat*}
+%
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%