From 90236d995dad33c141e24b86b31523eed76cd0b1 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Fri, 23 Dec 2022 21:17:49 +0100
Subject: [PATCH] Added Proximal Decoding: General Idea slides

---
 .../midterm/sections/decoding_algorithms.tex  | 75 +++++++++++++++++--
 1 file changed, 70 insertions(+), 5 deletions(-)

diff --git a/latex/presentations/midterm/sections/decoding_algorithms.tex b/latex/presentations/midterm/sections/decoding_algorithms.tex
index 510dec8..1717cc4 100644
--- a/latex/presentations/midterm/sections/decoding_algorithms.tex
+++ b/latex/presentations/midterm/sections/decoding_algorithms.tex
@@ -8,13 +8,78 @@
 
 \begin{frame}[t]
     \frametitle{Proximal Decoding: General Idea}
-    \todo{Settle on a convetion for titles: Use overview slides subtitles or something else?}
 
-    \todo{TODO}
+    \begin{itemize}
+        \item Code constraint polynomial:
+            \begin{align*}
+            h\left( \boldsymbol{x} \right) =
+                \underbrace{\sum_{j=1}^{n} \left( x_j^2 - 1 \right)^2}_{\text{Bipolar constraint}}
+                + \underbrace{\sum_{i=1}^{m} \left[ \left(
+                    \prod_{j\in\mathcal{A}\left( i \right)} x_j\right) -1 \right]^2}
+                        _{\text{Parity constraint}}
+            \end{align*}
+        \item Approximation of prior PDF:
+            \begin{align*}
+                f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)
+                    = \frac{1}{\left| \mathcal{C}\left( \boldsymbol{H} \right)  \right| }
+                        \sum_{c \in \mathcal{C}\left( \boldsymbol{H} \right) }
+                            \delta\left( \boldsymbol{x} - \boldsymbol{c} \right) 
+                    \approx \frac{1}{Z} e^{-\gamma h\left( x \right) }
+            \end{align*}
+            \todo{How does $\lim_{\gamma\rightarrow\infty}\left( \ldots \right)
+                = 1 = \delta\left( \ldots \right) $?}
+        \item MAP rule:
+            \begin{align*}
+                \hat{\boldsymbol{x}}
+                    = arg\max_{x\in\mathbb{R}}\left[
+                        f_{\boldsymbol{Y}}\left( \boldsymbol{y} | \boldsymbol{x} \right)
+                        f_{\boldsymbol{X}}\left( \boldsymbol{x} \right) \right]
+                    = arg\max_{x\in\mathbb{R}}\left[
+                        e^{-L\left( \boldsymbol{y} | \boldsymbol{x}\right)}
+                        e^{-\gamma h\left( \boldsymbol{x} \right) } \right]
+            \end{align*}
+    \end{itemize}
 \end{frame}
 
 \begin{frame}[t]
-    \frametitle{Algorithm}
+    \frametitle{Proximal Decoding: General Idea}
+
+    \begin{itemize}
+        \item Objective function:
+            \begin{align*}
+                f\left( \boldsymbol{x} \right)
+                    = L\left( \boldsymbol{y} | \boldsymbol{x} \right)
+                    + \gamma h\left( \boldsymbol{x} \right),
+                    \hspace{5mm} L\left( \boldsymbol{y} | \boldsymbol{x} \right)
+                        = - \ln\left( f_{\boldsymbol{Y}}
+                            \left( \boldsymbol{y} | \boldsymbol{x} \right)  \right)
+            \end{align*}
+           \todo{Note about explaning notational difference between $f$ and $f_X$ or $f_Y$}
+        \item Code proximal operator:
+            \begin{align*}
+                \text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv
+                    arg\min_{\boldsymbol{z}\in\mathbb{R}} \left(
+                        \gamma h\left( \boldsymbol{z} \right) + \frac{1}{2} \left| \boldsymbol{z}
+                            - \boldsymbol{x} \right|^2  \right)\\
+                        &\approx \boldsymbol{x} - \gamma \nabla h\left( \boldsymbol{x} \right),
+                        \hspace{5mm} \gamma \text{ small}
+            \end{align*}
+            \todo{Euclidean norm symbol}
+        \item Iterative decoding process:
+            \begin{align*}
+                \boldsymbol{r}^{\left( k+1 \right) } &= \boldsymbol{s}^{\left( k \right) }
+                    - \omega \nabla L\left( \boldsymbol{y} | \boldsymbol{s}^{\left( k \right) }
+                    \right), \hspace{5mm} \omega > 0
+                    \hspace{10mm} \text{``Gradient descent step''}\\
+                \boldsymbol{s}^{\left( k+1 \right) } &= \boldsymbol{r}^{\left( k+1 \right) }
+                    - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)
+                    \hspace{22mm} \text{``Code proximal step''}
+            \end{align*}
+    \end{itemize}
+\end{frame}
+
+\begin{frame}[t]
+    \frametitle{Proximal Decoding: Algorithm}
 
     \todo{TODO}
 \end{frame}
@@ -25,13 +90,13 @@
 \label{sub:Alg ADMM}
 
 \begin{frame}[t]
-    \frametitle{General Idea}
+    \frametitle{ADMM Decoding: General Idea}
 
     \todo{TODO}
 \end{frame}
 
 \begin{frame}[t]
-    \frametitle{Algorithm}
+    \frametitle{ADMM Decoding: Algorithm}
 
     \todo{TODO}
 \end{frame}