Complete first draft of warm-start sliding-window decoding section

src/thesis/chapters/4_decoding_under_dems.tex

@@ -724,21 +724,20 @@ structure by passing soft information from earlier to later spatial positions.
 
 % Passing messages requires messages
 
-% TODO: Move this to the intro?
 The act of passing messages from one window to the next requires
-there being messages at the end of decoding that are still relevant
-to the decoding process.
-This somewhat limits the variety of inner decoders the warm-start
+there being messages after completing decoding of one window that are
+still relevant to the decoding of the next.
+This may somewhat limit the variety of \emph{inner decoders}, i.e.,
+the decoders decoding the individual windows, the warm-start
 initialization can be used with.
 E.g., \ac{bp}+\ac{osd} does not immediately seem suitable, though
 this remains to be investigated.
-\red{[Something general about the available decoders]}
-\red{[Define inner decoder]}
+We chose to investigate first plain \ac{bp} due to its simplicity and
+then \ac{bpgd} because of the availability of recently computed messages.
 
-% Proposed modification: Mathematical definition
-
-\content{Mention that our own work ties into the bottom category in
-\Cref{fig:literature}}
+% TODO: Include this?
+% \content{Mention that our own work ties into the bottom category in
+% \Cref{fig:literature}}
 
 %%%%%%%%%%%%%%%%
 \subsection{Warm Start For Belief Propagation Decoding}
@@ -861,7 +860,7 @@ this remains to be investigated.
 
     \caption{
         \red{Visualization of the messages used for the
-        initialization of the next window.}
+        initialization of the next window under BP decoding.}
         \Acfp{vn} are represented using green circles while \acfp{cn}
         are represented using blue squares.
     }
@@ -877,12 +876,14 @@ we perform a \emph{warm start} by initializing the messages in the
 overlapping region to the values last held during the decoding of the
 previous window.
 
+% Practical realization: Problem with naive approach
+
 To see how we realize this in practice, we reiterate the steps of the
 \ac{bp} algorithm
 \begin{align}
     \label{eq:init}
     \text{Initialization: } & L_{i \rightarrow j} = \tilde{L}_i \\[3mm]
-    \text{\ac{cn} Update (Sum-Product): }&
+    \text{\ac{cn} Update (SPA): }&
     \displaystyle L_{i \leftarrow j} =
     2\cdot(-1)^{s_j}\cdot\tanh^{-1}
     \!\left(
@@ -891,8 +892,8 @@ To see how we realize this in practice, we reiterate the steps of the
     \right) \\[3mm]
     \text{\ac{cn} Update (Min-Sum): }&
     \displaystyle L_{i \leftarrow j} = (-1)^{s_j}\cdot \prod_{i'
-    \in \mathcal{N}(j)\setminus i} \sign \left( L_{i' \rightarrow j}
-    \right) \cdot \min_{i' \in \mathcal{N}(j)\setminus i} \lvert
+    \in \mathcal{N}(j)\setminus \{i\}} \sign \left( L_{i' \rightarrow j}
+    \right) \cdot \min_{i' \in \mathcal{N}(j)\setminus \{i\}} \lvert
     L_{i'\rightarrow j} \rvert \\[3mm]
     \label{eq:vn_update}
     \text{\ac{vn} Update: } & \displaystyle L_{i \rightarrow j} =
@@ -921,6 +922,8 @@ This means that simply initializing the edges in the overlap region
 with the exising $L_{i\rightarrow j}$ messages and starting the
 decoding of the next window with a \ac{cn} update is not enough.
 
+% Practical realization: working approach
+
 We can resolve this issue by initializing the edges using the existing
 \ac{cn} to \ac{vn} messages $L_{i\leftarrow j}$ and beginning the
 decoding of the next window with a \ac{vn} update instead.
@@ -928,6 +931,8 @@ This way, we recompute the existing $L_{i\rightarrow j}$ messages and
 additionally compute the messages crossing the window boundary.
 We can then continue decoding the next window as usual.
 
+% Practical realization: Simplification of algorithm
+
 We can further simplify the algorithm.
 Looking carefully at \Cref{eq:vn_update} we notice that when the
 \ac{cn} to \ac{vn} messages $L_{i\leftarrow j}$ have been zero-initialized,
@@ -954,7 +959,7 @@ Note that the decoding procedure performed on the individual windows
 % tex-fmt: off
 \tikzexternaldisable
 \begin{algorithm}[t]
-    \caption{Sliding-window belief propagation (BP) decoding algorithm with warm-start.}
+    \caption{Sliding-window belief propagation (BP) decoding algorithm with warm start.}
     \label{alg:warm_start_bp}
     \begin{algorithmic}[1]
         \State \textbf{Initialize:} $\hat{\bm{e}}^\text{total} \leftarrow \bm{0}$
@@ -988,14 +993,25 @@ Note that the decoding procedure performed on the individual windows
 \subsection{Warm Start for Belief Propagation with Guided Decimation Decoding}
 \label{subsec:Warm-Start Belief Propagation with Guided Decimation Decoding}
 
-% Intro
+% Intro: Recap of BPGD
 
-\content{Call back to previous paragraph: BPGD is one option where
-relevant messages are available}
-\content{Modified structure of BPGD $\rightarrow$ In addition to
-messages, pass decimation info}
-\content{(?) Explicitly mention decimation info = channel llrs?}
-\content{(?) Algorithm}
+We now direct our attention at using \ac{bpgd} as an inner decoder.
+Recall that for \ac{bpgd}, after a number $T \in \mathbb{N}$ of
+iterations we decimate
+the most reliable \ac{vn}, meaning we perform a hard decision and
+remove it from the following decoding process.
+
+This means that when moving from one window to the next, we now have
+more information available: not just the \ac{bp} messages but also the
+information about what \acp{vn} were decimated and to what values.
+We call this \emph{decimation information} in the following.
+We can extend \Cref{alg:warm_start_bp} by additionally passing the
+decimation information after initializing the \ac{cn} to \ac{vn} messages.
+\Cref{fig:messages_decimation_tanner} visualizes this process.
+
+% TODO: Do this in the fundamentals chapter. Then write a proper
+% algorithm for warm-start sliding-window decoding with BPGD as well
+%\content{(?) Explicitly mention decimation info = channel llrs?}
 
 \begin{figure}[t]
     \centering
@@ -1125,12 +1141,45 @@ messages, pass decimation info}
     \end{tikzpicture}
 
     \caption{
-        Visualization of the messages and decimation information used for the
-        initialization of the next window.
+        \red{Visualization of the messages and decimation information
+            used for the
+        initialization of the next window under \ac{bpgd} decoding}.
+        \Acfp{vn} are represented using green circles while \acfp{cn}
+        are represented using blue squares.
     }
     \label{fig:messages_decimation_tanner}
 \end{figure}
 
+% % tex-fmt: off
+% \tikzexternaldisable
+% \begin{algorithm}[t]
+%     \caption{Sliding-window decoding algorithm with warm start for generic inner decoder.}
+%     \label{alg:warm_start_general}
+%     \begin{algorithmic}[1]
+%         \State \textbf{Initialize:} $\hat{\bm{e}}^\text{total} \leftarrow \bm{0}$
+%         \State \textbf{Initialize:} $L_{i\leftarrow j} = 0
+%             ~\forall~ i\in \mathcal{I}, j\in \mathcal{J}$
+%         \For{$\ell = 0, \ldots, n_\text{win}-1$}
+%             \State Obtain $\hat{\bm{e}}^{(\ell)}$ from inner decoder
+%             \State $\displaystyle\left(\hat{\bm{e}}^\text{total}\right)_{\mathcal{I}^{(\ell)}_\text{commit}} \leftarrow \hat{\bm{e}}^{(\ell)}_\text{commit}$
+%             \State $\displaystyle\left(\bm{s}\right)_{\mathcal{J}_\text{overlap}^{(\ell)}}
+%                 \leftarrow \bm{H}_\text{overlap}^{(\ell)}
+%                 \left( \hat{\bm{e}}_\text{commit}^{(\ell)} \right)^\text{T}$
+%             \If{$\ell < n_\text{win} - 1$}
+%                 \State $L^{(\ell+1)}_{i\leftarrow j} \leftarrow
+%                 L^{(\ell)}_{i\leftarrow j}
+%                 ~\forall~ i \in \mathcal{I}_\text{overlap}^{(\ell)},
+%                     j \in \mathcal{J}_\text{overlap}^{(\ell)}$
+%             \EndIf
+%         \EndFor
+%         \State \textbf{return} $\hat{\bm{e}}$
+%     \end{algorithmic}
+% \end{algorithm}
+% \tikzexternalenable
+% % tex-fmt: on
+%
+% \content{Make algorithm 4 specific to BPGD?}
+
 \section{Numerical Results}
 \label{sec:Numerical Results}