520 lines
22 KiB
TeX
520 lines
22 KiB
TeX
\chapter{Comparison of Proximal Decoding and \acs{LP} Decoding using \acs{ADMM}}%
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\label{chapter:comparison}
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In this chapter, proximal decoding and \ac{LP} Decoding using \ac{ADMM} are compared.
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First the two algorithms are compared on a theoretical basis.
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Subsequently, their respective simulation results are examined, and their
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differences are interpreted on the basis of their theoretical structure.
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%some similarities between the proximal decoding algorithm
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%and \ac{LP} decoding using \ac{ADMM} are be pointed out.
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%The two algorithms are compared and their different computational and decoding
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%performance is interpreted on the basis of their theoretical structure.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theoretical Comparison}%
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\label{sec:comp:theo}
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\ac{ADMM} and the proximal gradient method can both be expressed in terms of
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proximal operators \cite[Sec. 4.4]{proximal_algorithms}.
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When using \ac{ADMM} as an optimization method to solve the \ac{LP} decoding
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problem specifically, this is not quite possible because of the multiple
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constraints.
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In spite of that, the two algorithms still show some striking similarities.
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To see the first of these similarities, the \ac{LP} decoding problem in
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equation (\ref{eq:lp:relaxed_formulation}) can be slightly rewritten using the
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\textit{indicator functions} $g_j : \mathbb{R}^{d_j} \rightarrow
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\left\{ 0, +\infty \right\} \hspace{1mm}, j\in\mathcal{J}$ for the polytopes
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$\mathcal{P}_{d_j}, \hspace{1mm} j\in\mathcal{J}$, defined as%
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%
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\begin{align*}
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g_j\left( \boldsymbol{t} \right) := \begin{cases}
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0, & \boldsymbol{t} \in \mathcal{P}_{d_j} \\
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+\infty, & \boldsymbol{t} \not\in \mathcal{P}_{d_j}
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\end{cases}
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,\end{align*}
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%
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by moving the constraints into the objective function, as shown in figure
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\ref{fig:ana:theo_comp_alg:admm}.
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Both algorithms are composed of an iterative approach consisting of two
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alternating steps.
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The objective functions of both problems are similar in that they
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both comprise two parts: one associated to the likelihood that a given
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codeword was sent and one associated to the constraints the decoding process
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is subjected to.
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%
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\begin{figure}[h]
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\centering
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\begin{subfigure}{0.42\textwidth}
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\centering
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\begin{align*}
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\text{minimize}\hspace{2mm} & \underbrace{L\left( \boldsymbol{y} \mid
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\tilde{\boldsymbol{x}} \right)}_{\text{Likelihood}}
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+ \underbrace{\gamma h\left( \tilde{\boldsymbol{x}} \right)}
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_{\text{Constraints}} \\
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\text{subject to}\hspace{2mm} &\tilde{\boldsymbol{x}} \in \mathbb{R}^n
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\end{align*}
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\begin{genericAlgorithm}[caption={}, label={},
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basicstyle=\fontsize{10}{18}\selectfont
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]
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Initialize $\boldsymbol{r}, \boldsymbol{s}, \omega, \gamma$
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while stopping critierion unfulfilled do
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$\boldsymbol{r} \leftarrow \boldsymbol{r}
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+ \omega \nabla L\left( \boldsymbol{y} \mid \boldsymbol{s} \right) $
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$\boldsymbol{s} \leftarrow
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\textbf{prox}_{\scaleto{\gamma h}{7.5pt}}\left( \boldsymbol{r} \right) $|\Suppressnumber|
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|\Reactivatenumber|
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end while
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return $\boldsymbol{s}$
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\end{genericAlgorithm}
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\caption{Proximal decoding}
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\label{fig:ana:theo_comp_alg:prox}
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\end{subfigure}\hfill%
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\begin{subfigure}{0.55\textwidth}
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\centering
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\begin{align*}
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\text{minimize}\hspace{5mm} &
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\underbrace{\boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}}
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_{\text{Likelihood}}
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+ \underbrace{\sum\nolimits_{j\in\mathcal{J}} g_j\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}} \right) }
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_{\text{Constraints}} \\
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\text{subject to}\hspace{5mm} &
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\tilde{\boldsymbol{c}} \in \left[ 0, 1 \right]^n
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\end{align*}
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\begin{genericAlgorithm}[caption={}, label={},
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basicstyle=\fontsize{10}{18}\selectfont
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]
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Initialize $\tilde{\boldsymbol{c}}, \boldsymbol{z}, \boldsymbol{u}, \boldsymbol{\gamma}, \rho$
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while stopping criterion unfulfilled do
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$\tilde{\boldsymbol{c}} \leftarrow \argmin_{\tilde{\boldsymbol{c}}}
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\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
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+ \frac{\rho}{2}\sum_{j\in\mathcal{J}} \left\Vert
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\boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j
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+ \boldsymbol{u}_j \right\Vert \right)$
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$\boldsymbol{z}_j \leftarrow \textbf{prox}_{g_j}
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\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
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+ \boldsymbol{u}_j \right), \hspace{5mm}\forall j\in\mathcal{J}$
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$\boldsymbol{u}_j \leftarrow \boldsymbol{u}_j
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+ \tilde{\boldsymbol{c}} - \boldsymbol{z}_j, \hspace{15.25mm}\forall j\in\mathcal{J}$
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end while
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return $\tilde{\boldsymbol{c}}$
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\end{genericAlgorithm}
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\caption{LP decoding using \ac{ADMM}}
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\label{fig:ana:theo_comp_alg:admm}
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\end{subfigure}%
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\caption{Comparison of the proximal gradient method and \ac{ADMM}}
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\label{fig:ana:theo_comp_alg}
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\end{figure}%
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%
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Their major difference is that while with proximal decoding the constraints
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are regarded in a global context, considering all parity checks at the same
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time, with \ac{ADMM} each parity check is
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considered separately and in a more local context (line 4 in both algorithms).
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This difference means that while with proximal decoding the alternating
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minimization of the two parts of the objective function inevitably leads to
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oscillatory behavior (as explained in section
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\ref{subsec:prox:conv_properties}), this is not the case with \ac{ADMM}, which
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partly explains the disparate decoding performance of the two methods.
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Furthermore, while with proximal decoding the step considering the constraints
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is realized using gradient descent - amounting to an approximation -
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with \ac{ADMM} it reduces to a number of projections onto the parity polytopes
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$\mathcal{P}_{d_j}$ which always provide exact results.
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The contrasting treatment of the constraints (global and approximate with
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proximal decoding as opposed to local and exact with \ac{LP} decoding using
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\ac{ADMM}) also leads to different prospects when the decoding process gets
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stuck in a local minimum.
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With proximal decoding this occurs due to the approximate nature of the
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calculation, whereas with \ac{LP} decoding it occurs due to the approximate
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formulation of the constraints - independent of the optimization method
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itself.
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The advantage which arises because of this when employing \ac{LP} decoding is
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that it can be easily detected \todo{Not 'easily' detected}, when the algorithm gets stuck - it
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returns a solution corresponding to a pseudocodeword, the components of which
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are fractional.
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Moreover, when a valid codeword is returned, it is also the \ac{ML} codeword.
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This means that additional redundant parity-checks can be added successively
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until the codeword returned is valid and thus the \ac{ML} solution is found
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\cite[Sec. IV.]{alp}.
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In terms of time complexity, the two decoding algorithms are comparable.
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Each of the operations required for proximal decoding can be performed
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in linear time for \ac{LDPC} codes (see section \ref{subsec:prox:comp_perf}).
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The same is true for the $\tilde{\boldsymbol{c}}$- and $\boldsymbol{u}$-update
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steps of \ac{LP} decoding using \ac{ADMM}, while
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the projection step has a worst-case time complexity of
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$\mathcal{O}\left( n^2 \right)$ and an average complexity of
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$\mathcal{O}\left( n \right)$ (see section TODO, \cite[Sec. VIII.]{lautern}).
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Both algorithms can be understood as message-passing algorithms, \ac{LP}
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decoding using \ac{ADMM} as similarly to \cite[Sec. III. D.]{original_admm}
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or \cite[Sec. II. B.]{efficient_lp_dec_admm} and proximal decoding by
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starting with algorithm \ref{alg:prox},
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substituting for the gradient of the code-constraint polynomial and separating
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it into two parts.
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The algorithms in their message-passing form are depicted in figure
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\ref{fig:comp:message_passing}.
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$M_{j\to i}$ denotes a message transmitted from \ac{CN} j to \ac{VN} i.
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$M_{j\to}$ signifies the special case where a \ac{VN} transmits the same
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message to all \acp{VN}.
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%
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\begin{figure}[h]
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\centering
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\begin{subfigure}{0.48\textwidth}
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\centering
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\begin{genericAlgorithm}[caption={}, label={},
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% basicstyle=\fontsize{10}{16}\selectfont
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]
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Initialize $\boldsymbol{r}, \boldsymbol{s}, \omega, \gamma$
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while stopping critierion unfulfilled do
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for j in $\mathcal{J}$ do
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$p_j \leftarrow \prod_{i\in N_c\left( j \right) } r_i $
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$M_{j\to} \leftarrow p_j^2 - p_j$|\Suppressnumber|
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|\vspace{0.22mm}\Reactivatenumber|
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end for
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for i in $\mathcal{I}$ do
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$s_i \leftarrow s_i + \gamma \left[ 4\left( s_i^2 - 1 \right)s_i
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\phantom{\frac{4}{s_i}}\right.$|\Suppressnumber|
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|\Reactivatenumber|$\left.+ \frac{4}{s_i}\sum_{j\in N_v\left( i \right) }
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M_{j\to} \right] $
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$r_i \leftarrow r_i + \omega \left( s_i - y_i \right)$
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end for
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end while
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return $\boldsymbol{s}$
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\end{genericAlgorithm}
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\caption{Proximal decoding}
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\label{fig:comp:message_passing:proximal}
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\end{subfigure}%
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\hfill
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\begin{subfigure}{0.48\textwidth}
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\centering
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\begin{genericAlgorithm}[caption={}, label={},
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% basicstyle=\fontsize{10}{16}\selectfont
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]
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Initialize $\tilde{\boldsymbol{c}}, \boldsymbol{z}, \boldsymbol{u}, \boldsymbol{\gamma}, \rho$
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while stopping criterion unfulfilled do
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for j in $\mathcal{J}$ do
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$\boldsymbol{z}_j \leftarrow \Pi_{P_{d_j}}\left(
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\boldsymbol{T}_j\tilde{\boldsymbol{c}} + \boldsymbol{u}_j\right)$
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$\boldsymbol{u}_j \leftarrow \boldsymbol{u}_j + \boldsymbol{T}_j\tilde{\boldsymbol{c}}
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- \boldsymbol{z}_j$
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$M_{j\to i} \leftarrow \left( z_j \right)_i - \left( u_j \right)_i,
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\hspace{3mm} \forall i \in N_c\left( j \right) $
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end for
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for i in $\mathcal{I}$ do
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$\tilde{c}_i \leftarrow \frac{1}{d_i}
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\left(\sum_{j\in N_v\left( i \right) } M_{j\to i}
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- \frac{\gamma_i}{\mu} \right)$|\Suppressnumber|
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|\vspace{7mm}\Reactivatenumber|
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end for
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end while
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return $\tilde{\boldsymbol{c}}$
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\end{genericAlgorithm}
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\caption{\ac{LP} decoding using \ac{ADMM}}
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\label{fig:comp:message_passing:admm}
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\end{subfigure}%
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\caption{The proximal gradient method and \ac{LP} decoding using \ac{ADMM}
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as message passing algorithms}
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\label{fig:comp:message_passing}
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\end{figure}%
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%
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It is evident that while the two algorithms are very similar in their general
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structure, with \ac{LP} decoding using \ac{ADMM}, multiple messages have to be
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computed for each check node (line 6 in figure
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\ref{fig:comp:message_passing:admm}), whereas
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with proximal decoding, the same message is transmitted to all \acp{VN}
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(line 5 of figure \ref{fig:comp:message_passing:proximal}).
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This means that while both algorithms have an average time complexity of
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$\mathcal{O}\left( n \right)$, more arithmetic operations are required in the
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\ac{ADMM} case.
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In conclusion, the two algorithms have a very similar structure, where the
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parts of the objective function relating to the likelihood and to the
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constraints are minimized in an alternating fashion.
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With proximal decoding this minimization is performed for all constraints at once
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in an approximative manner, while with \ac{LP} decoding using \ac{ADMM} it is
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performed for each constraint individually and with exact results.
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In terms of time complexity, both algorithms are, on average, linear with
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respect to $n$, although for \ac{LP} decoding using \ac{ADMM} significantly
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more arithmetic operations are necessary in each iteration.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Comparison of Simulation Results}%
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\label{sec:comp:res}
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\begin{itemize}
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\item The comparison of actual implementations is always debatable /
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contentious, since it is difficult to separate differences in
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algorithm performance from differences in implementation
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\item No large difference in computational performance $\rightarrow$
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Parallelism cannot come to fruition as decoding is performed on the
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same number of cores for both algorithms (Multiple decodings in parallel)
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\item Nonetheless, in realtime applications / applications where the focus
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is not the mass decoding of raw data, \ac{ADMM} has advantages, since
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the decoding of a single codeword is performed faster
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\item \ac{ADMM} faster than proximal decoding $\rightarrow$
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Parallelism
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\item Proximal decoding faster than \ac{ADMM} $\rightarrow$ dafuq
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(larger number of iterations before convergence? More values to compute for ADMM?)
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\end{itemize}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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width=\textwidth,
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height=0.75\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_963965.csv};
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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%{res/hybrid/2d_ber_fer_dfr_963965.csv};
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{res/admm/ber_2d_963965.csv};
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\addplot[PineGreen, line width=1pt, mark=triangle]
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table [col sep=comma, x=SNR, y=FER,]
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{res/generic/fer_ml_9633965.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=96, k=48$
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\cite[\text{96.3.965}]{mackay_enc}}
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\end{subfigure}%
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\hfill%
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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width=\textwidth,
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height=0.75\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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{res/admm/ber_2d_bch_31_26.csv};
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\addplot[PineGreen, line width=1pt, mark=triangle*]
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table [x=SNR, y=FER, col sep=comma,
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discard if gt={SNR}{5.5},
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discard if lt={SNR}{1},
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]
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{res/generic/fer_ml_bch_31_26.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{BCH code with $n=31, k=26$}
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\end{subfigure}%
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\vspace{3mm}
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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width=\textwidth,
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height=0.75\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.05},
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discard if gt={SNR}{5.5}]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={mu}{3.0},
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discard if gt={SNR}{5.5}]
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{res/admm/ber_2d_20433484.csv};
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\addplot[PineGreen, line width=1pt, mark=triangle, solid]
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table [col sep=comma, x=SNR, y=FER,
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discard if gt={SNR}{5.5}]
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{res/generic/fer_ml_20433484.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=204, k=102$
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\cite[\text{204.33.484}]{mackay_enc}}
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\end{subfigure}%
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\hfill%
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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|
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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width=\textwidth,
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height=0.75\textwidth,
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]
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_20455187.csv};
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\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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{res/admm/ber_2d_20455187.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 5, 10 \right)$-regular \ac{LDPC} code with $n=204, k=102$
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\cite[\text{204.55.187}]{mackay_enc}}
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\end{subfigure}%
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\vspace{3mm}
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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|
|
|
\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={FER},
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ymode=log,
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ymax=1.5, ymin=8e-5,
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width=\textwidth,
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height=0.75\textwidth,
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]
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|
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\addplot[RedOrange, line width=1pt, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_40833844.csv};
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|
\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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|
table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
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{res/admm/ber_2d_40833844.csv};
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\end{axis}
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|
\end{tikzpicture}
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|
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|
\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=204, k=102$
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|
\cite[\text{204.33.484}]{mackay_enc}}
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|
\end{subfigure}%
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|
\hfill%
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|
\begin{subfigure}[t]{0.48\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
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|
\begin{axis}[
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|
grid=both,
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|
xlabel={$E_b / N_0$}, ylabel={FER},
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|
ymode=log,
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|
ymax=1.5, ymin=8e-5,
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|
width=\textwidth,
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|
height=0.75\textwidth,
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|
]
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|
|
|
\addplot[RedOrange, line width=1pt, mark=*, solid]
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|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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|
{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
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|
table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
|
|
{res/admm/ber_2d_pegreg252x504.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{LDPC code (progressive edge growth construction) with $n=504, k=252$
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|
\cite[\text{PEGReg252x504}]{mackay_enc}}
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|
\end{subfigure}%
|
|
|
|
\vspace{5mm}
|
|
|
|
\begin{subfigure}[t]{\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[hide axis,
|
|
xmin=10, xmax=50,
|
|
ymin=0, ymax=0.4,
|
|
legend columns=1,
|
|
legend style={draw=white!15!black}]
|
|
\addlegendimage{RedOrange, line width=1pt, mark=*, solid}
|
|
\addlegendentry{Proximal decoding}
|
|
|
|
\addlegendimage{NavyBlue, line width=1pt, mark=triangle, densely dashed}
|
|
\addlegendentry{\acs{LP} decoding using \acs{ADMM}}
|
|
|
|
\addlegendimage{PineGreen, line width=1pt, mark=triangle*, solid}
|
|
\addlegendentry{\acs{ML} decoding}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{subfigure}
|
|
|
|
\caption{Comparison of decoding performance between proximal decoding and \ac{LP} decoding
|
|
using \ac{ADMM}}
|
|
\label{fig:comp:prox_admm_dec}
|
|
\end{figure}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[grid=both,
|
|
xlabel={$n$}, ylabel={Time per frame (s)},
|
|
width=0.6\textwidth,
|
|
height=0.45\textwidth,
|
|
legend style={at={(0.5,-0.42)},anchor=south},
|
|
legend cell align={left},]
|
|
|
|
\addplot[RedOrange, only marks, mark=*]
|
|
table [col sep=comma, x=n, y=spf]
|
|
{res/proximal/fps_vs_n.csv};
|
|
\addlegendentry{Proximal decoding}
|
|
\addplot[PineGreen, only marks, mark=triangle*]
|
|
table [col sep=comma, x=n, y=spf]
|
|
{res/admm/fps_vs_n.csv};
|
|
\addlegendentry{\acs{LP} decoding using \acs{ADMM}}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Timing requirements of the proximal decoding imlementation%
|
|
\protect\footnotemark{}}
|
|
\label{fig:comp:time}
|
|
\end{figure}%
|
|
%
|
|
\footnotetext{asdf}
|
|
%
|