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Added Proximal Decoding: General Idea slides

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Andreas Tsouchlos 2022-12-23 21:17:49 +01:00
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latex/presentations/midterm/sections/decoding_algorithms.tex
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@ -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}