169 lines
7.2 KiB
TeX
169 lines
7.2 KiB
TeX
\chapter{Analysis of Results}%
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\label{chapter:Analysis of Results}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{LP Decoding using ADMM}%
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\label{sec:ana:LP Decoding using ADMM}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Proximal Decoding}%
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\label{sec:ana:Proximal Decoding}
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\begin{itemize}
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\item Parameter choice
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\item FER
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\item Improved implementation
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Comparison of BP, Proximal Decoding and LP Decoding using ADMM}%
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\label{sec:ana:Comparison of BP, Proximal Decoding and LP Decoding using ADMM}
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\begin{itemize}
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\item Decoding performance
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\item Complexity \& runtime(mention difficulty in reaching conclusive
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results when comparing implementations)
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theoretical Comparison of Proximal Decoding and LP Decoding using ADMM}%
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\label{sec:Theoretical Comparison of Proximal Decoding and LP Decoding using ADMM}
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In this section, 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 explained on the basis of their theoretical structure.
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\ac{ADMM} and the proximal gradient method can both be expressed in terms of
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proximal operators.
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They are both composed of an iterative approach consisting of two
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alternating steps.
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In both cases each step minimizes one distinct part of the objective function.
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The approaches they are based on, however, are fundamentally different.
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In figure \ref{fig:ana:theo_comp_alg} the two algorithms are juxtaposed,
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in conjuction with the optimization problems they are meant to solve, in their
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proximal operator form.%
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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{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{11}{17}\selectfont
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]
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Initialize variables
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while stopping critierion not satisfied 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 gradient method}
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\label{fig:ana:theo_comp_alg:prox}
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\end{subfigure}\hfill%
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\begin{subfigure}{0.48\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_{j\in\mathcal{J}} g_j\left(
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\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 \mathbb{R}^n
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% \boldsymbol{T}_j\tilde{\boldsymbol{c}} = \boldsymbol{z}_j\hspace{3mm}
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% \forall j\in\mathcal{J}
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\end{align*}
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\begin{genericAlgorithm}[caption={}, label={},
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basicstyle=\fontsize{11}{17}\selectfont
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]
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Initialize variables
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while stopping criterion not satisfied do
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$\tilde{\boldsymbol{c}} \leftarrow \textbf{prox}_{
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\scaleto{\nu \cdot \boldsymbol{\gamma}^{\text{T}}\tilde{\boldsymbol{c}}}{8.5pt}}
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\left( \boldsymbol{z} - \boldsymbol{u} \right) $
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$\boldsymbol{z}_j \leftarrow \textbf{prox}_{\scaleto{g_j}{7pt}}
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\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
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+ \boldsymbol{T}_j\boldsymbol{u} \right) \hspace{5mm}\forall j\in\mathcal{J}$
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$\boldsymbol{u} \leftarrow \boldsymbol{u}
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+ \tilde{\boldsymbol{c}} - \boldsymbol{z}$
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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{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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\todo{Show how $\tilde{\boldsymbol{c}} \leftarrow \textbf{prox}
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_{1 / \mu \cdot \boldsymbol{\gamma}^{\text{T}}\tilde{\boldsymbol{c}}}
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\left( \boldsymbol{z} - \boldsymbol{u} \right) $
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is the same as
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$\boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
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+ \sum_{j\in\mathcal{J}} \boldsymbol{\lambda}^\text{T}_j
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\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j \right)
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+ \frac{\mu}{2}\sum_{j\in\mathcal{J}}
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\lVert \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j \rVert^2_2$}%
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%
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\noindent 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 codeword is
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subjected to.
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Their major difference is that the two parts of the objective minimized with
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proximal decoding are both functions of the same variable
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$\tilde{\boldsymbol{x}}$, whereas with \ac{ADMM} the two parts are functions
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of different variables: $\tilde{\boldsymbol{c}}$ and $\boldsymbol{z}_{[1:m]}$.
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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 behaviour (as explained in section (TODO)), this is not the
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case with \ac{ADMM}.
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Another aspect partly explaining the disparate decoding performance is the
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difference in the minimization step handling the constraints.
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While with proximal decoding it is performed using gradient
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descent - amounting to an approximation - with \ac{ADMM} it reduces to a
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number of projections onto the parity polytopes $\mathcal{P}_{d_j}$ - which
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always provide exact results.
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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?)
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\end{itemize}
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