\section{Decoding Algorithms}%
\label{sec:Decoding Algorithms}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Proximal Decoding}%
\label{sub:Alg Proximal Decoding}

\begin{frame}[t]
    \frametitle{Proximal Decoding: General Idea \cite{proximal_paper}}

    \begin{itemize}
        \item MAP rule:
            \begin{align*}
                \hat{\boldsymbol{x}}
                    = \argmax_{x\in\mathbb{R}}
                        f_{\boldsymbol{Y}}\left( \boldsymbol{y} | \boldsymbol{x} \right)
                        f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)
                    = \argmax_{x\in\mathbb{R}}
                        e^{-L\left( \boldsymbol{y} | \boldsymbol{x}\right)}
                        f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)
            \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} - \left( -1 \right)^{\boldsymbol{c}} \right) 
                    \approx \frac{1}{Z} e^{-\gamma h\left( x \right) }
            \end{align*}
        \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}},
            \hspace{5mm}\mathcal{A}\left( i \right) \equiv \left\{
                j | j\in \mathcal{J},
                \boldsymbol{H}_{j,i} = 1
            \right\},
            i \in \mathcal{I}
            \end{align*}
    \end{itemize}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
    \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*}
            \note{Notational difference between $f$ and $f_X$ or $f_Y$}
        \item Code proximal operator \cite{proximal_algorithms}:
            \begin{align*}
                \text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv
                    \argmin_{\boldsymbol{z}\in\mathbb{R}} \left(
                        \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*}
        \item Iterative decoding process:
            \begin{align*}
                \boldsymbol{r} &\leftarrow \boldsymbol{s}
                    - \omega \nabla L\left( \boldsymbol{y} | \boldsymbol{s}
                    \right), \hspace{5mm} \omega > 0
                    \hspace{10mm} \text{``Gradient descent step''}\\
                \boldsymbol{s} &\leftarrow \boldsymbol{r}
                    - \gamma \nabla h\left( \boldsymbol{r}
                    \right) \hspace{29mm} \text{``Code proximal step''}
            \end{align*}
    \end{itemize}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t, fragile]
    \frametitle{Proximal Decoding: Algorithm}
    \begin{itemize}
        \item Resulting iterative decoding algorithm \cite{proximal_paper}:
    \end{itemize}
            
    \vspace{2mm}
    \begin{algorithm}[caption={}, label={}]
$\boldsymbol{s} \leftarrow \boldsymbol{0}$
for $K$ iterations do
    $\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \nabla L \left( \boldsymbol{y} \mid \boldsymbol{s} \right) $
    $\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) $
    $\boldsymbol{\hat{x}} \leftarrow \text{sign}\left( \boldsymbol{s} \right) $
    If $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$, then return $\boldsymbol{\hat{c}}$
end for
return $\boldsymbol{\hat{c}}$
    \end{algorithm}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{LP Decoding}%
\label{sub:LP Decoding}

\begin{frame}[t]
    \frametitle{LP Decoding \cite{feldman_paper}}

    \begin{minipage}[c]{0.6\linewidth}
        \begin{itemize}
            \item Codeword polytope:
                \begin{align*}
                    \text{poly}\left( \mathcal{C} \right) =
                        \left\{
                            \sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}}
                                \boldsymbol{c} : \lambda_{\boldsymbol{c}} \ge  0,
                            \sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}} = 1
                        \right\},
                        \hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R}
                \end{align*}
            \item Cost function:
                \begin{align*}
                    \sum_{i=1}^{n} \gamma_i c_i,
                    \hspace{5mm}\gamma_i = \log\left(
                        \frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
                \end{align*}
            \item LP formulation of ML decoding:
                \begin{align*}
                    &\text{minimize } \sum_{i=1}^{n} \gamma_i f_i \\
                    &\text{subject to } \boldsymbol{f}\in\text{poly}\left( \mathcal{C} \right) 
                \end{align*}
        \end{itemize}
    \end{minipage}%
    \hfill%
    \begin{minipage}[c]{0.4\linewidth}
        \begin{figure}[H]
            \centering

            \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
                                    draw, circle, inner sep=0pt, minimum size=4pt]

            \tdplotsetmaincoords{60}{245}
            \begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
                % Cube

                \draw[dashed] (0, 0, 0) -- (2, 0, 0);
                \draw[dashed] (2, 0, 0) -- (2, 0, 2);
                \draw[] (2, 0, 2) -- (0, 0, 2);
                \draw[] (0, 0, 2) -- (0, 0, 0);

                \draw[] (0, 2, 0) -- (2, 2, 0);
                \draw[] (2, 2, 0) -- (2, 2, 2);
                \draw[] (2, 2, 2) -- (0, 2, 2);
                \draw[] (0, 2, 2) -- (0, 2, 0);

                \draw[] (0, 0, 0) -- (0, 2, 0);
                \draw[dashed] (2, 0, 0) -- (2, 2, 0);
                \draw[] (2, 0, 2) -- (2, 2, 2);
                \draw[] (0, 0, 2) -- (0, 2, 2);

                % Codeword Polytope

                \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 0, 2);
                \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 2, 0);
                \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (0, 2, 2);

                \draw[line width=1pt, color=KITblue] (2, 0, 2) -- (2, 2, 0);
                \draw[line width=1pt, color=KITblue] (2, 0, 2) -- (0, 2, 2);

                \draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);

                % Polytope Annotations

                \node[codeword] (c000) at (0, 0, 0) {};% {$\left( 0, 0, 0 \right) $};
                \node[codeword] (c101) at (2, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
                \node[codeword] (c110) at (2, 2, 0) {};% {$\left( 1, 1, 0 \right) $};
                \node[codeword] (c011) at (0, 2, 2) {};% {$\left( 0, 1, 1 \right) $};

                \node[color=KITblue, right=0cm of c000] {$\left( 0, 0, 0 \right) $};
                \node[color=KITblue, above=0cm of c101] {$\left( 1, 0, 1 \right) $};
                \node[color=KITblue, left=0cm of c110] {$\left( 1, 1, 0 \right) $};
                \node[color=KITblue, left=0cm of c011] {$\left( 0, 1, 1 \right) $};

                % f

                \node[color=KITgreen, fill=KITgreen,
                      draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.7, 0.7, 1) {};
                \node[color=KITgreen, right=0cm of f] {$\boldsymbol{f}$};
            \end{tikzpicture}
            \caption{$\text{poly}\left( \mathcal{C} \right)$ for $n=3$}
        \end{figure}
    \end{minipage}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[t]
%    \frametitle{LP Relaxation}
%
%    \begin{minipage}[c]{0.6\linewidth}
%        \begin{itemize}
%            \item Set of all variable nodes incident to a check node:
%                \begin{align*}
%                    N\left( j \right) \equiv \left\{
%                        i | i\in \mathcal{I},
%                        \boldsymbol{H}_{j,i} = 1
%                    \right\},
%                    j \in \mathcal{J}
%                \end{align*}
%                \begin{align*}
%                    S \subseteq N\left( j \right), \left| S \right| \text{odd}
%                \end{align*}
%            \item Relaxed polytope representation:
%                \begin{align*}
%                    \sum_{i\in \left( N\left( j \right) \setminus S\right) } f_i
%                        + \sum_{i\in S} \left( 1 - f_i \right) \ge 1
%                \end{align*}
%                ``$\boldsymbol{f}$ is separated by at least one bitflip
%                from all illegal configurations''
%        \end{itemize}
%    \end{minipage}%
%    \hfill%
%    \begin{minipage}[c]{0.4\linewidth}
%        \begin{figure}[H]
%            \centering
%
%            \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
%                                    draw, circle, inner sep=0pt, minimum size=4pt]
%
%            \tdplotsetmaincoords{60}{245}
%            \begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
%                % Cube
%
%                \draw[dashed] (0, 0, 0) -- (2, 0, 0);
%                \draw[dashed] (2, 0, 0) -- (2, 0, 2);
%                \draw[] (2, 0, 2) -- (0, 0, 2);
%                \draw[] (0, 0, 2) -- (0, 0, 0);
%
%                \draw[] (0, 2, 0) -- (2, 2, 0);
%                \draw[] (2, 2, 0) -- (2, 2, 2);
%                \draw[] (2, 2, 2) -- (0, 2, 2);
%                \draw[] (0, 2, 2) -- (0, 2, 0);
%
%                \draw[] (0, 0, 0) -- (0, 2, 0);
%                \draw[dashed] (2, 0, 0) -- (2, 2, 0);
%                \draw[] (2, 0, 2) -- (2, 2, 2);
%                \draw[] (0, 0, 2) -- (0, 2, 2);
%
%                % Codeword Polytope
%
%                \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 0, 2);
%                \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 2, 0);
%                \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (0, 2, 2);
%
%                \draw[line width=1pt, color=KITblue] (2, 0, 2) -- (2, 2, 0);
%                \draw[line width=1pt, color=KITblue] (2, 0, 2) -- (0, 2, 2);
%
%                \draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);
%
%                % Polytope Annotations
%
%                \node[codeword, color=KITred] (c111) at (2, 2, 2) {};% {$\left( 0, 0, 0 \right) $};
%                \node[codeword, color=KITred] (c001) at (0, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
%                \node[codeword, color=KITred] (c100) at (2, 0, 0) {};% {$\left( 1, 1, 0 \right) $};
%                \node[codeword, color=KITred] (c010) at (0, 2, 0) {};% {$\left( 0, 1, 1 \right) $};
%
%                \node[color=KITred, left=0cm of c111] {$\left( 1, 1, 1 \right) $};
%                \node[color=KITred, right=0cm of c001] {$\left( 0, 0, 1 \right) $};
%                \node[color=KITred, right=0.35cm of c100] {$\left( 1, 0, 0 \right) $};
%                \node[color=KITred, below=0cm of c010] {$\left( 0, 1, 0 \right) $};
%            \end{tikzpicture}
%            \caption{Relaxed polytope for $n=3$}
%        \end{figure}
%    \end{minipage}
%    \todo{How is this a relaxation and not just an alternative formulation?
%        We have just switched out valid codewords for invalid ones}
%    \todo{Is LP Relaxation relevant as theoretical background?}
%\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{ADMM}%
%\label{sub:Alg ADMM}
%
%\begin{frame}[t]
%    \frametitle{ADMM Decoding: General Idea}
%
%    \todo{TODO}
%\end{frame}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[t]
%    \frametitle{ADMM Decoding: Algorithm}
%
%    \todo{TODO}
%\end{frame}

