From 52ba8c67eeb4b3f0e91bb11b2fb7d5c200e5de61 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Sun, 12 Mar 2023 20:23:22 +0100
Subject: [PATCH] Added footnote about discrete -> continuous; Added quotation
 marks; minor other changes

---
 latex/thesis/abbreviations.tex                |  5 +++++
 latex/thesis/chapters/decoding_techniques.tex | 22 +++++++++++--------
 2 files changed, 18 insertions(+), 9 deletions(-)

diff --git a/latex/thesis/abbreviations.tex b/latex/thesis/abbreviations.tex
index abe0b43..8aae37c 100644
--- a/latex/thesis/abbreviations.tex
+++ b/latex/thesis/abbreviations.tex
@@ -97,6 +97,11 @@
     long  = probability density function
 }
 
+\DeclareAcronym{PMF}{
+    short = PMF,
+    long  = probability mass function
+}
+
 %
 % V
 %
diff --git a/latex/thesis/chapters/decoding_techniques.tex b/latex/thesis/chapters/decoding_techniques.tex
index 9b6ad37..452e3f8 100644
--- a/latex/thesis/chapters/decoding_techniques.tex
+++ b/latex/thesis/chapters/decoding_techniques.tex
@@ -25,14 +25,14 @@ available optimization algorithms.
 Generally, the original decoding problem considered is either the \ac{MAP} or
 the \ac{ML} decoding problem:%
 %
-\begin{align*}
+\begin{align}
     \hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
     p_{\boldsymbol{C} \mid \boldsymbol{Y}} \left(\boldsymbol{c} \mid \boldsymbol{y}
-        \right)\\
+        \right) \label{eq:dec:map}\\
     \hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
     f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c}
-        \right) 
-.\end{align*}%
+        \right) \label{eq:dec:ml}
+.\end{align}%
 %
 The goal is to arrive at a formulation, where a certain objective function
 $g : \mathbb{R}^n \rightarrow \mathbb{R}^n $ must be minimized under certain constraints:%
@@ -707,7 +707,11 @@ non-convex optimization formulation of the \ac{MAP} decoding problem.
 
 In order to derive the objective function, the authors begin with the
 \ac{MAP} decoding rule, expressed as a continuous maximization problem%
-\footnote{The }%
+\footnote{The expansion of the domain to be continuous doesn't constitute a
+material difference.
+The only change is that what previously were \acp{PMF} now have to be expressed
+in terms of \acp{PDF}}
+over $\boldsymbol{x}$
 :%
 %
 \begin{align}
@@ -726,7 +730,7 @@ The likelihood $f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
 determined by the channel model.
 The prior \ac{PDF} $f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$ is also
 known, as the equal probability assumption is made on
-$\mathcal{C}\left( \boldsymbol{H} \right)$.
+$\mathcal{C}$.
 However, since the considered domain is continuous,
 the prior \ac{PDF} cannot be ignored as a constant during the minimization
 as is often done, and has a rather unwieldy representation:%
@@ -843,14 +847,14 @@ The second step thus becomes%
     \hspace{5mm}\gamma > 0,\text{ small}
 .\end{align*}
 %
-While the approximation of the prior \ac{PDF} made in \ref{eq:prox:prior_pdf_approx}
+While the approximation of the prior \ac{PDF} made in equation (\ref{eq:prox:prior_pdf_approx})
 theoretically becomes better
 with larger $\gamma$, the constraint that $\gamma$ be small is important,
 as it keeps the effect of $h\left( \boldsymbol{x} \right) $ on the landscape
 of the objective function small.
 Otherwise, unwanted stationary points, including local minima, are introduced.
-The authors say that in practice, the value of $\gamma$ should be adjusted
-according to the decoding performance \cite[Sec. 3.1]{proximal_paper}.
+The authors say that ``in practice, the value of $\gamma$ should be adjusted
+according to the decoding performance.'' \cite[Sec. 3.1]{proximal_paper}.
 
 %The components of the gradient of the code-constraint polynomial can be computed as follows:%
 %%