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src/thesis/chapters/4_decoding_under_dems.tex
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@@ -2035,6 +2035,9 @@ For the underlying \ac{bp} step we use the \ac{spa} variant rather
than the min-sum approximation employed in
\Cref{subsec:Belief Propagation}, since this made the implementation
of the guided decimation more straightforward.
Furthermore, we set $T=1$, as this eases the
computational requirements and \cite{yao_belief_2024} showed that most of
the gain can be achieved even for low values of $T$.
\begin{figure}[t]
\centering
@@ -2518,8 +2521,8 @@ iterations can change the outcome, which is why each cold-start curve
reaches a flat plateau.
The warm-start curves exhibit the same two regimes, but with the
opposite outcome in the second one, which is exactly what the
hypothesis from the previous paragraph predicts.
opposite outcome in the second one, which is exactly what our earlier
hypothesis predicts.
At low $n_\text{iter}$, decimation has not yet taken hold and the
warm-start initialization carries forward only the \ac{bp} messages
in any meaningful sense, so the warm-start variant outperforms its
@@ -2540,6 +2543,17 @@ decisions of the \acp{vn}.
We do not have a definitive explanation for the roughness visible in some
of the warm-start curves and limit ourselves to noting it.
% [Thread] Turn to previous way of warm-start
The natural consequence of the previous diagnosis is to drop the
problematic part of the warm-start initialization for \ac{bpgd} and
to carry over only the \ac{bp} messages on the edges of the overlap
region, as in \Cref{fig:messages_tanner}, while leaving the channel
\acp{llr} of the next window in their original cold-start state.
Note that some information about the previous window's decimation
state is still implicitly carried over through the \ac{bp} messages,
since the decimation decisions were made based on the messages themselves.
\begin{figure}[t]
\centering
\hspace*{-6mm}
@@ -2610,6 +2624,7 @@ of the warm-start curves and limit ourselves to noting it.
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\label{fig:bpgd_msg_W}
\end{subfigure}%
\hfill%
\begin{subfigure}{0.5\textwidth}
@@ -2680,13 +2695,71 @@ of the warm-start curves and limit ourselves to noting it.
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\label{fig:bpgd_msg_F}
\end{subfigure}
\caption{
\red{\lipsum[2]}
}
\label{fig:bpgd_msg}
\end{figure}
% [Experimental parameters] Figure 4.12
\Cref{fig:bpgd_msg} repeats the experiment of \Cref{fig:bpgd_wf}
with the modified warm-start procedure that carries over only the
\ac{bp} messages.
All other experimental parameters are unchanged: the maximum number
of inner \ac{bp} iterations is $n_\text{iter} = 5000$, and the
physical error rate is swept from $p = 0.001$ to $p = 0.004$ in steps
of $0.0005$.
The cold-start curves (dashed) are identical to those in
\Cref{fig:bpgd_wf}.
The warm-start curves are shown with solid lines.
\Cref{fig:bpgd_msg_W} sweeps over the window size with
$W \in \{3, 4, 5\}$ at fixed step size $F = 1$, and
\Cref{fig:bpgd_msg_F} sweeps over the step size with
$F \in \{1, 2, 3\}$ at fixed window size $W = 5$.
% [Description] Figure 4.12
The warm-start curves now lie below their cold-start counterparts
across both panels and across the entire physical error rate range,
in contrast to \Cref{fig:bpgd_wf}.
In \Cref{fig:bpgd_msg_W}, larger window sizes again yield lower
per-round \acp{ler} for both warm- and cold-start, and the warm-start
advantage over cold-start is more pronounced for $W \in \{4, 5\}$
than for $W = 3$, where the warm- and cold-start curves nearly coincide.
In \Cref{fig:bpgd_msg_F}, smaller step sizes again yield lower
per-round \acp{ler} for both warm- and cold-start, and the warm-start
advantage over cold-start is most pronounced for $F = 1$ and shrinks
as $F$ grows.
% [Description] Interpretation 4.12
Removing the channel \acp{llr} from the warm-start initialization lifts
the warm-start regression observed in \Cref{fig:bpgd_wf},
and warm-start now consistently outperforms cold-start.
The dependence on the window size and the step size also recovers
the qualitative behavior we observed for plain \ac{bp} in
\Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}: a larger overlap
between consecutive windows, achieved either by enlarging $W$ or by
decreasing $F$, both improves the absolute decoding performance and
increases the warm-start advantage over cold-start.
This is consistent with the original effective-iterations picture.
Without the premature hard decisions from carried-over decimation
information, the warm-start initialization once again amounts to
additional \ac{bp} iterations on the \acp{vn} of the overlap region,
and the larger the overlap, the more such effective iterations are gained.
% [Thread] As before, view max iter behavior
Finally, we repeat the iteration-budget sweep of \Cref{fig:bpgd_iter}
with the message-only warm-start procedure.
This serves both to verify that the premature hard decision effect
does not reappear at any iteration count and to compare the warm- and
cold-start curves across the entire range of $n_\text{iter}$ available to us.
\begin{figure}[t]
\centering
\hspace*{-6mm}
@@ -2759,6 +2832,7 @@ of the warm-start curves and limit ourselves to noting it.
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\label{fig:bpgd_msg_iter_W}
\end{subfigure}%
\hfill%
\begin{subfigure}{0.48\textwidth}
@@ -2831,10 +2905,74 @@ of the warm-start curves and limit ourselves to noting it.
\caption{\red{Lorem ipsum dolor sit amet, consectetur adipiscing
elit, sed do eiusmod tempor incididunt}}
\label{fig:bpgd_msg_iter_F}
\end{subfigure}
\caption{
\red{\lipsum[2]}
}
\label{fig:bpgd_msg_iter}
\end{figure}
% [Experimental parameters] Figure 4.13
\Cref{fig:bpgd_msg_iter} repeats the experiment of
\Cref{fig:bpgd_iter} with the modified warm-start procedure that
carries over only the \ac{bp} messages.
All other experimental parameters are unchanged: the physical error
rate is fixed at $p = 0.0025$ and the iteration budget is swept over
$n_\text{iter} \in \{32, 128, 256, 512, 1024, 1536, 2048, 2560,
3072, 3584, 4096\}$.
The cold-start curves (dashed) are identical to those in
\Cref{fig:bpgd_iter}.
\Cref{fig:bpgd_msg_iter_W} sweeps over the window size with
$W \in \{3, 4, 5\}$ at fixed step size $F = 1$, and
\Cref{fig:bpgd_msg_iter_F} sweeps over the step size with
$F \in \{1, 2, 3\}$ at fixed window size $W = 5$.
% [Description] Figure 4.13
The warm-start curves now again lie consistently below their cold-start
counterparts across both panels and across the entire range of
$n_\text{iter}$, contrary to \Cref{fig:bpgd_iter}.
The warm-start curves furthermore track the overall shape of the
corresponding cold-start curves closely, including the iteration
count at which they drop sharply and the level at which they plateau.
The warm-start improvement over cold-start grows with the window size
in \Cref{fig:bpgd_msg_iter_W} and shrinks with the step size in
\Cref{fig:bpgd_msg_iter_F}, with the largest gap visible at $W = 5$
and at $F = 1$, respectively.
% [Interpretation] Figure 4.13
These observations match our expectations.
With only the \ac{bp} messages carried over, the warm-start
initialization no longer freezes any \acp{vn} in the next window
The dependence of this benefit on $W$ and $F$ also recovers the
pattern observed for plain \ac{bp} in
\Cref{fig:whole_vs_cold_vs_warm,fig:bp_f_over_p}:
larger overlap, achieved by larger $W$ or smaller $F$, yields more
effective extra iterations and therefore a larger warm-start gain.
% BPGD conclusion
We conclude our investigation into the performance of warm-start
sliding-window decoding under \ac{bpgd} by summarizing our findings.
Warm-starting the inner decoder still provides a consistent
performance gain when the inner decoder is upgraded from plain
\ac{bp} to its guided-decimation variant, but only if some care is
taken in choosing what to carry over.
Passing the channel \acp{llr} along with the \ac{bp} messages,
as suggested by naively carrying over the warm-start idea to \ac{bpgd},
leads to premature hard decisions on \acp{vn} in the overlap region.
This leads to warm-start initialization actually worsening the
performance compared to cold-start initialization.
Restricting the warm start to the \ac{bp} messages alone removes
this effect and recovers a consistent warm-start improvement over
cold-start that follows the same behavior as for plain \ac{bp} with
regard to overlap.
A second observation specific to \ac{bpgd} is that its iteration
requirements are substantially larger than those of plain \ac{bp}:
the per-round \ac{ler} drops sharply only once the iteration budget
is on the order of the number of \acp{vn} in each window.