Added proximal decoding algorithm

latex/presentations/midterm/sections/decoding_algorithms.tex

@@ -59,12 +59,11 @@
             \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)\\
+                        \gamma h\left( \boldsymbol{z} \right) + \frac{1}{2} \lVert \boldsymbol{z}
+                            - \boldsymbol{x} \rVert^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) }
@@ -78,10 +77,19 @@
     \end{itemize}
 \end{frame}
 
-\begin{frame}[t]
+\begin{frame}[t, fragile]
     \frametitle{Proximal Decoding: Algorithm}
-
-    \todo{TODO}
+        \begin{algorithm}[caption={}, label={}]
+$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$
+for $k=0$ to $K-1$ do
+    $\boldsymbol{r}^{\left( k+1 \right)} = \boldsymbol{s}^{(k)} - \omega \nabla L \left( \boldsymbol{s}^{(k)}; \boldsymbol{y} \right) $
+    Compute $\nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)$
+    $\boldsymbol{s}^{\left( k+1 \right)} = \boldsymbol{r}^{(k+1)} - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right) $
+    $\boldsymbol{\hat{x}} = \text{sign}\left( \boldsymbol{s}^{\left( k+1 \right) } \right) $
+    If $\boldsymbol{\hat{x}}$ passes the parity check condition, break the loop.
+end for
+Output $\boldsymbol{\hat{x}}$
+        \end{algorithm}
 \end{frame}