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Replace \boldsymbol by \bm

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Andreas Tsouchlos 2024-01-18 21:56:00 +01:00
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@ -9,6 +9,7 @@
\usepackage{dsfont} \usepackage{dsfont}
\usepackage{mleftright} \usepackage{mleftright}
\usepackage{bbm} \usepackage{bbm}
\usepackage{bm}
\usepackage[ \usepackage[
backend=biber, backend=biber,
style=ieee, style=ieee,
@ -201,43 +202,43 @@ codewords, the lengths of which are denoted by $k \in \mathbb{N}$
and $n \in \mathbb{N}$, respectively, with $k \le n$. and $n \in \mathbb{N}$, respectively, with $k \le n$.
The set of codewords $\mathcal{C} \subset \mathbb{F}_2^n$ of a binary linear The set of codewords $\mathcal{C} \subset \mathbb{F}_2^n$ of a binary linear
code can be represented using the parity-check matrix code can be represented using the parity-check matrix
$\boldsymbol{H} \in \mathbb{F}_2^{m \times n} $, where $m$ represents the $\bm{H} \in \mathbb{F}_2^{m \times n} $, where $m$ represents the
number of parity-checks: number of parity-checks:
% %
\begin{align*} \begin{align*}
\mathcal{C} := \left\{ \boldsymbol{c} \in \mathbb{F}_2^n : \mathcal{C} := \left\{ \bm{c} \in \mathbb{F}_2^n :
\boldsymbol{H}\boldsymbol{c}^\text{T} = \boldsymbol{0} \right\} \bm{H}\bm{c}^\text{T} = \bm{0} \right\}
\end{align*} \end{align*}
% %
The check nodes $j \in \mathcal{J}:=\left\{1, \ldots, m\right\}$ each The check nodes $j \in \mathcal{J}:=\left\{1, \ldots, m\right\}$ each
correspond to a parity check, i.e., a row of $\boldsymbol{H}$. correspond to a parity check, i.e., a row of $\bm{H}$.
The variable nodes $i \in \mathcal{I}:=\left\{1, \ldots, n\right\}$ correspond The variable nodes $i \in \mathcal{I}:=\left\{1, \ldots, n\right\}$ correspond
to the components of a codeword being subjected to a parity check, i.e., to the components of a codeword being subjected to a parity check, i.e.,
to the columns of $\boldsymbol{H}$. to the columns of $\bm{H}$.
The neighborhood of a parity check $j$, i.e., the set of indices of components The neighborhood of a parity check $j$, i.e., the set of indices of components
relevant for the according parity check, is denoted by relevant for the according parity check, is denoted by
$\mathcal{N}_c(j) := \left\{i \in \mathcal{I}: \boldsymbol{H}\negthinspace_{j,i} = 1 \right\}, $\mathcal{N}_c(j) := \left\{i \in \mathcal{I}: \bm{H}\negthinspace_{j,i} = 1 \right\},
\hspace{2mm} j \in \mathcal{J}$. \hspace{2mm} j \in \mathcal{J}$.
In order to transmit a codeword $\boldsymbol{c} \in \mathbb{F}_2^n$, it is In order to transmit a codeword $\bm{c} \in \mathbb{F}_2^n$, it is
mapped onto a \textit{binary phase shift keying} (BPSK) symbol via mapped onto a \textit{binary phase shift keying} (BPSK) symbol via
$\boldsymbol{x} = 1 - 2\boldsymbol{c}$, with $\bm{x} = 1 - 2\bm{c}$, with
$ \boldsymbol{x} \in \left\{\pm 1\right\}^n$, which is then transmitted over an $ \bm{x} \in \left\{\pm 1\right\}^n$, which is then transmitted over an
AWGN channel. AWGN channel.
The received vector $\boldsymbol{y} \in \mathbb{R}^n$ is decoded to obtain an The received vector $\bm{y} \in \mathbb{R}^n$ is decoded to obtain an
estimate of the transmitted codeword, denoted as estimate of the transmitted codeword, denoted as
$\hat{\boldsymbol{c}} \in \mathbb{F}_2^n$. $\hat{\bm{c}} \in \mathbb{F}_2^n$.
A distinction is made between $\boldsymbol{x} \in \left\{\pm 1\right\}^n$ A distinction is made between $\bm{x} \in \left\{\pm 1\right\}^n$
and $\tilde{\boldsymbol{x}} \in \mathbb{R}^n$, and $\tilde{\bm{x}} \in \mathbb{R}^n$,
the former denoting the BPSK symbol physically transmitted over the channel and the former denoting the BPSK symbol physically transmitted over the channel and
the latter being used as a variable during the optimization process. the latter being used as a variable during the optimization process.
The posterior probability of having transmitted $\boldsymbol{x}$ when receiving The posterior probability of having transmitted $\bm{x}$ when receiving
$\boldsymbol{y}$ is expressed as a \textit{probability mass function} (PMF) $\bm{y}$ is expressed as a \textit{probability mass function} (PMF)
$P_{\boldsymbol{X}\mid\boldsymbol{Y}}(\boldsymbol{x} \mid \boldsymbol{y})$. $P_{\bm{X}\mid\bm{Y}}(\bm{x} \mid \bm{y})$.
Likewise, the likelihood of receiving $\boldsymbol{y}$ upon transmitting Likewise, the likelihood of receiving $\bm{y}$ upon transmitting
$\boldsymbol{x}$ is expressed as a \textit{probability density function} (PDF) $\bm{x}$ is expressed as a \textit{probability density function} (PDF)
$f_{\boldsymbol{Y}\mid\boldsymbol{X}}(\boldsymbol{y} \mid \boldsymbol{x})$. $f_{\bm{Y}\mid\bm{X}}(\bm{y} \mid \bm{x})$.
%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%
@ -258,7 +259,7 @@ One such expression, formulated under the assumption of BPSK, is the
\textit{code-constraint polynomial} \cite{proximal_paper} \textit{code-constraint polynomial} \cite{proximal_paper}
% %
\begin{align*} \begin{align*}
h( \tilde{\boldsymbol{x}} ) = h( \tilde{\bm{x}} ) =
\underbrace{\sum_{i=1}^{n} \underbrace{\sum_{i=1}^{n}
\left( \tilde{x}_i^2-1 \right) ^2}_{\text{Bipolar constraint}} \left( \tilde{x}_i^2-1 \right) ^2}_{\text{Bipolar constraint}}
+ \underbrace{\sum_{j=1}^{m} \left[ + \underbrace{\sum_{j=1}^{m} \left[
@ -274,45 +275,45 @@ information regarding the code.
The channel model can be considered using the negative log-likelihood The channel model can be considered using the negative log-likelihood
% %
\begin{align*} \begin{align*}
L \mleft( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \mright) = -\ln\mleft( L \mleft( \bm{y} \mid \tilde{\bm{x}} \mright) = -\ln\mleft(
f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}} \mleft( f_{\bm{Y} \mid \tilde{\bm{X}}} \mleft(
\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \mright) \mright) \bm{y} \mid \tilde{\bm{x}} \mright) \mright)
.\end{align*} .\end{align*}
% %
The information about the channel and the code are consolidated in the objective The information about the channel and the code are consolidated in the objective
function \cite{proximal_paper} function \cite{proximal_paper}
% %
\begin{align*} \begin{align*}
g \mleft( \tilde{\boldsymbol{x}} \mright) g \mleft( \tilde{\bm{x}} \mright)
= L \mleft( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \mright) = L \mleft( \bm{y} \mid \tilde{\bm{x}} \mright)
+ \gamma h\mleft( \tilde{\boldsymbol{x}} \mright), + \gamma h\mleft( \tilde{\bm{x}} \mright),
\hspace{5mm} \gamma > 0% \hspace{5mm} \gamma > 0%
.\end{align*} .\end{align*}
% %
The objective function is minimized using the proximal gradient method, which The objective function is minimized using the proximal gradient method, which
amounts to iteratively performing two gradient-descent steps \cite{proximal_paper} amounts to iteratively performing two gradient-descent steps \cite{proximal_paper}
with the given objective function and considering AWGN channels. with the given objective function and considering AWGN channels.
To this end, two helper variables, $\boldsymbol{r}$ and $\boldsymbol{s}$, are To this end, two helper variables, $\bm{r}$ and $\bm{s}$, are
introduced, describing the result of each of the two steps: introduced, describing the result of each of the two steps:
% %
\begin{alignat}{3} \begin{alignat}{3}
\boldsymbol{r} &\leftarrow \boldsymbol{s} \bm{r} &\leftarrow \bm{s}
- \omega \mleft( \boldsymbol{s} - \boldsymbol{y} \mright) - \omega \mleft( \bm{s} - \bm{y} \mright)
\hspace{5mm }&&\omega > 0 \label{eq:r_update}\\ \hspace{5mm }&&\omega > 0 \label{eq:r_update}\\
\boldsymbol{s} &\leftarrow \boldsymbol{r} \bm{s} &\leftarrow \bm{r}
- \gamma \nabla h\mleft( \boldsymbol{r} \mright), - \gamma \nabla h\mleft( \bm{r} \mright),
\hspace{5mm} &&\gamma > 0 \label{eq:s_update} \hspace{5mm} &&\gamma > 0 \label{eq:s_update}
.\end{alignat} .\end{alignat}
% %
An equation for determining $\nabla h(\boldsymbol{r})$ is given in An equation for determining $\nabla h(\bm{r})$ is given in
\cite{proximal_paper}. \cite{proximal_paper}.
It should be noted that the variables $\boldsymbol{r}$ and $\boldsymbol{s}$ It should be noted that the variables $\bm{r}$ and $\bm{s}$
represent $\tilde{\boldsymbol{x}}$ during different represent $\tilde{\bm{x}}$ during different
stages of the decoding process. stages of the decoding process.
As the gradient of the code-constraint polynomial can attain very large values As the gradient of the code-constraint polynomial can attain very large values
in some cases, an additional step is introduced to ensure numerical stability: in some cases, an additional step is introduced to ensure numerical stability:
every current estimate $\boldsymbol{s}$ is projected onto every current estimate $\bm{s}$ is projected onto
$\left[-\eta, \eta\right]^n$ by a projection $\left[-\eta, \eta\right]^n$ by a projection
$\Pi_\eta : \mathbb{R}^n \rightarrow \left[-\eta, \eta\right]^n$, where $\eta$ $\Pi_\eta : \mathbb{R}^n \rightarrow \left[-\eta, \eta\right]^n$, where $\eta$
is a positive constant slightly larger than one, e.g., $\eta = 1.5$. is a positive constant slightly larger than one, e.g., $\eta = 1.5$.
@ -324,16 +325,16 @@ presented in Algorithm \ref{alg:proximal_decoding}.
\label{alg:proximal_decoding} \label{alg:proximal_decoding}
\begin{algorithmic} \begin{algorithmic}
\STATE $\boldsymbol{s} \leftarrow \boldsymbol{0}$ \STATE $\bm{s} \leftarrow \bm{0}$
\STATE \textbf{for} $K$ iterations \textbf{do} \STATE \textbf{for} $K$ iterations \textbf{do}
\STATE \hspace{5mm} $\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \left( \boldsymbol{s} - \boldsymbol{y} \right) $ \STATE \hspace{5mm} $\bm{r} \leftarrow \bm{s} - \omega \left( \bm{s} - \bm{y} \right) $
\STATE \hspace{5mm} $\boldsymbol{s} \leftarrow \Pi_\eta \left(\boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) \right)$ \STATE \hspace{5mm} $\bm{s} \leftarrow \Pi_\eta \left(\bm{r} - \gamma \nabla h\left( \bm{r} \right) \right)$
\STATE \hspace{5mm} $\boldsymbol{\hat{c}} \leftarrow \mathbbm{1}_{\left\{ \boldsymbol{s} \le 0 \right\}}$ \STATE \hspace{5mm} $\bm{\hat{c}} \leftarrow \mathbbm{1}_{\left\{ \bm{s} \le 0 \right\}}$
\STATE \hspace{5mm} \textbf{if} $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ \textbf{do} \STATE \hspace{5mm} \textbf{if} $\bm{H}\bm{\hat{c}} = \bm{0}$ \textbf{do}
\STATE \hspace{10mm} \textbf{return} $\boldsymbol{\hat{c}}$ \STATE \hspace{10mm} \textbf{return} $\bm{\hat{c}}$
\STATE \hspace{5mm} \textbf{end if} \STATE \hspace{5mm} \textbf{end if}
\STATE \textbf{end for} \STATE \textbf{end for}
\STATE \textbf{return} $\boldsymbol{\hat{c}}$ \STATE \textbf{return} $\bm{\hat{c}}$
\end{algorithmic} \end{algorithmic}
\end{algorithm} \end{algorithm}
@ -481,8 +482,8 @@ optimization process.%
at={(0.9775,0.97)},anchor=north east}] at={(0.9775,0.97)},anchor=north east}]
\addlegendimage{mark=none} \addlegendimage{mark=none}
\addlegendentry{ \addlegendentry{
$\nabla L\left(\boldsymbol{y} $\nabla L\left(\bm{y}
\mid \tilde{\boldsymbol{x}}\right)$ \mid \tilde{\bm{x}}\right)$
}; };
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
@ -530,22 +531,22 @@ optimization process.%
empty legend, empty legend,
at={(0.9775,0.97)},anchor=north east}] at={(0.9775,0.97)},anchor=north east}]
\addlegendimage{mark=none} \addlegendimage{mark=none}
\addlegendentry{$\nabla h\left(\tilde{\boldsymbol{x}}\right)$}; \addlegendentry{$\nabla h\left(\tilde{\bm{x}}\right)$};
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\fi \fi
\caption{Gradients \caption{Gradients
$\nabla L\left(\boldsymbol{y} \mid \tilde{\boldsymbol{x}}\right)$ $\nabla L\left(\bm{y} \mid \tilde{\bm{x}}\right)$
and $\nabla h \left( \tilde{\boldsymbol{x}} \right)$ for a repetition and $\nabla h \left( \tilde{\bm{x}} \right)$ for a repetition
code with $n=2$. code with $n=2$.
Shown for $\boldsymbol{y} = \begin{bmatrix} -0.5 & 0.8 \end{bmatrix}$. Shown for $\bm{y} = \begin{bmatrix} -0.5 & 0.8 \end{bmatrix}$.
} }
\label{fig:grad} \label{fig:grad}
\end{figure}% \end{figure}%
% %
In Fig. \ref{fig:prox:convergence_large_n}, we consider only component In Fig. \ref{fig:prox:convergence_large_n}, we consider only component
$\left(\tilde{\boldsymbol{x}}\right)_1$ of the estimate during a $\left(\tilde{\bm{x}}\right)_1$ of the estimate during a
decoding operation for the LDPC code used also for Fig. 1. decoding operation for the LDPC code used also for Fig. 1.
Two qualities may be observed. Two qualities may be observed.
First, we observe the average absolute values of the two gradients are equal, First, we observe the average absolute values of the two gradients are equal,
@ -585,14 +586,14 @@ oscillate after a certain number of iterations.%
table [col sep=comma, x=k, y=grad_h_0, table [col sep=comma, x=k, y=grad_h_0,
discard if gt={k}{300}] discard if gt={k}{300}]
{res/extreme_components_20433484_combined.csv}; {res/extreme_components_20433484_combined.csv};
\addlegendentry{$\left(\tilde{\boldsymbol{x}}\right)_1$} \addlegendentry{$\left(\tilde{\bm{x}}\right)_1$}
\addlegendentry{$\left(\nabla L\right)_1$} \addlegendentry{$\left(\nabla L\right)_1$}
\addlegendentry{$\left(\nabla h\right)_1$} \addlegendentry{$\left(\nabla h\right)_1$}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\fi \fi
\caption{Visualization of component $\left(\tilde{\boldsymbol{x}}\right)_1$ \caption{Visualization of component $\left(\tilde{\bm{x}}\right)_1$
for a decoding operation for a (3,6) regular LDPC code with for a decoding operation for a (3,6) regular LDPC code with
$n=204, k=102$ \cite[\text{204.33.484}]{mackay}. $n=204, k=102$ \cite[\text{204.33.484}]{mackay}.
Parameters used for simulation: $\gamma = 0.05, \omega = 0.05, Parameters used for simulation: $\gamma = 0.05, \omega = 0.05,
@ -608,7 +609,7 @@ Considering the magnitude of the oscillation of the gradient of the code constra
polynomial, some interesting behavior may be observed. polynomial, some interesting behavior may be observed.
Fig. \ref{fig:p_error} shows the probability that a component of the estimate Fig. \ref{fig:p_error} shows the probability that a component of the estimate
is wrong, determined through a Monte Carlo simulation, when the components of is wrong, determined through a Monte Carlo simulation, when the components of
$\boldsymbol{c}$ are ordered from smallest to largest oscillation of $\bm{c}$ are ordered from smallest to largest oscillation of
$\left(\nabla h\right)_i$. $\left(\nabla h\right)_i$.
The lower the magnitude of the oscillation, the higher the probability that the The lower the magnitude of the oscillation, the higher the probability that the
@ -640,7 +641,7 @@ the probability that a given component was decoded incorrectly.%
\fi \fi
\caption{Probability that a component of the estimated codeword \caption{Probability that a component of the estimated codeword
$\hat{\boldsymbol{c}}\in \mathbb{F}_2^n$ is erroneous for a (3,6) regular $\hat{\bm{c}}\in \mathbb{F}_2^n$ is erroneous for a (3,6) regular
LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay}. LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay}.
The indices $i'$ are ordered such that the amplitude of oscillation of The indices $i'$ are ordered such that the amplitude of oscillation of
$\left(\nabla h\right)_{i'}$ increases with $i'$. $\left(\nabla h\right)_{i'}$ increases with $i'$.
@ -656,7 +657,7 @@ If a valid codeword has been reached, i.e., if the algorithm has converged,
we return this solution. we return this solution.
Otherwise, $N \in \mathbb{N}$ components are selected based on the criterion Otherwise, $N \in \mathbb{N}$ components are selected based on the criterion
presented above. presented above.
Beginning with the recent estimate $\hat{\boldsymbol{c}} \in \mathbb{F}_2^n$, Beginning with the recent estimate $\hat{\bm{c}} \in \mathbb{F}_2^n$,
all variations of words with the selected components modified are then all variations of words with the selected components modified are then
generated and an ``ML-in-the-list'' step is performed. generated and an ``ML-in-the-list'' step is performed.
@ -665,13 +666,13 @@ generated and an ``ML-in-the-list'' step is performed.
\label{alg:ml-in-the-list} \label{alg:ml-in-the-list}
\begin{algorithmic} \begin{algorithmic}
\STATE Find valid codewords under $\left(\hat{\boldsymbol{c}}_{l}\right)_{1=1}^{2^N}$ \STATE Find valid codewords under $\left(\hat{\bm{c}}_{l}\right)_{1=1}^{2^N}$
\STATE \textbf{if} no valid codewords exist \STATE \textbf{if} no valid codewords exist
\STATE \hspace{5mm} Compute $\langle \hat{\boldsymbol{c}}_l, \hat{\boldsymbol{c}} \rangle$ for all variations $\boldsymbol{c}_l$ \STATE \hspace{5mm} Compute $\langle \hat{\bm{c}}_l, \hat{\bm{c}} \rangle$ for all variations $\bm{c}_l$
\STATE \textbf{else} \STATE \textbf{else}
\STATE \hspace{5mm} Compute $\langle \hat{\boldsymbol{c}}_l, \hat{\boldsymbol{c}} \rangle$ for valid codewords \STATE \hspace{5mm} Compute $\langle \hat{\bm{c}}_l, \hat{\bm{c}} \rangle$ for valid codewords
\STATE \textbf{end if} \STATE \textbf{end if}
\STATE \textbf{return} $\hat{\boldsymbol{c}}_l$ with highest $\langle \hat{\boldsymbol{c}}_l, \hat{\boldsymbol{c}} \rangle$ \STATE \textbf{return} $\hat{\bm{c}}_l$ with highest $\langle \hat{\bm{c}}_l, \hat{\bm{c}} \rangle$
\end{algorithmic} \end{algorithmic}
\end{algorithm}% \end{algorithm}%
% %
@ -681,18 +682,18 @@ generated and an ``ML-in-the-list'' step is performed.
\label{alg:improved} \label{alg:improved}
\begin{algorithmic} \begin{algorithmic}
\STATE $\boldsymbol{s} \leftarrow \boldsymbol{0}$ \STATE $\bm{s} \leftarrow \bm{0}$
\STATE \textbf{for} $K$ iterations \textbf{do} \STATE \textbf{for} $K$ iterations \textbf{do}
\STATE \hspace{5mm} $\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \left( \boldsymbol{s} - \boldsymbol{y} \right) $ \STATE \hspace{5mm} $\bm{r} \leftarrow \bm{s} - \omega \left( \bm{s} - \bm{y} \right) $
\STATE \hspace{5mm} $\boldsymbol{s} \leftarrow \Pi_\eta \left(\boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) \right)$ \STATE \hspace{5mm} $\bm{s} \leftarrow \Pi_\eta \left(\bm{r} - \gamma \nabla h\left( \bm{r} \right) \right)$
\STATE \hspace{5mm} $\boldsymbol{\hat{c}} \leftarrow \mathds{1} \left\{ \text{sign}\left( \boldsymbol{s} \right) = -1 \right\}$ \STATE \hspace{5mm} $\bm{\hat{c}} \leftarrow \mathds{1} \left\{ \text{sign}\left( \bm{s} \right) = -1 \right\}$
\STATE \hspace{10mm} \textbf{if} $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ \textbf{do} \STATE \hspace{10mm} \textbf{if} $\bm{H}\bm{\hat{c}} = \bm{0}$ \textbf{do}
\STATE \hspace{10mm} \textbf{return} $\boldsymbol{\hat{c}}$ \STATE \hspace{10mm} \textbf{return} $\bm{\hat{c}}$
\STATE \hspace{5mm} \textbf{end if} \STATE \hspace{5mm} \textbf{end if}
\STATE \textbf{end for} \STATE \textbf{end for}
\STATE $\textcolor{KITblue}{\text{Estimate $N$ wrong bit indices $\mathcal{I} = \{i_1,\ldots,i_N\}$}}$ \STATE $\textcolor{KITblue}{\text{Estimate $N$ wrong bit indices $\mathcal{I} = \{i_1,\ldots,i_N\}$}}$
\STATE $\textcolor{KITblue}{\text{Generate candidate list $\left(\hat{\boldsymbol{c}}_{l}\right)_{l=1}^{2^N}$ by varying bits in $\mathcal{I}$}}$\vspace{1mm} \STATE $\textcolor{KITblue}{\text{Generate candidate list $\left(\hat{\bm{c}}_{l}\right)_{l=1}^{2^N}$ by varying bits in $\mathcal{I}$}}$\vspace{1mm}
\STATE $\textcolor{KITblue}{\textbf{return ml\textunderscore in\textunderscore the\textunderscore list}\left(\left(\hat{\boldsymbol{c}}_l\right)_{1=1}^{2^N}\right)}$ \STATE $\textcolor{KITblue}{\textbf{return ml\textunderscore in\textunderscore the\textunderscore list}\left(\left(\hat{\bm{c}}_l\right)_{1=1}^{2^N}\right)}$
\end{algorithmic} \end{algorithmic}
\end{algorithm} \end{algorithm}