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latex/thesis/chapters/appendix.tex
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latex/thesis/chapters/appendix.tex
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\appendix
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\chapter{\acs{LP} Decoding using \acs{ADMM} as a Proximal Algorithm}%
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\label{chapter:LD Decoding using ADMM as a Proximal Algorithm}
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\todo{Find out how to properly title and section appendix}
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%For problems of the form%
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%\begin{align*}
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% \text{minimize}\hspace{2mm} & f\left( \boldsymbol{x} \right)
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% + g\left( \boldsymbol{A}\boldsymbol{x} \right) \\
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% \text{subject to}\hspace{2mm} & \boldsymbol{x} \in \mathbb{R}^n
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%,\end{align*}%
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%%
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%a version of \ac{ADMM}, \textit{linearized \ac{ADMM}}, can be expressed
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%as a proximal algorithm \cite[Sec. 4.4.2]{proximal_algorithms}:
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%%
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%\begin{align*}
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% \boldsymbol{x} &\leftarrow \textbf{prox}_{\mu f}\left( \boldsymbol{x}
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% - \frac{\mu}{\lambda}\boldsymbol{A}^\text{T}\left( \boldsymbol{A}\boldsymbol{x}
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% - \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
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% \boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left( \boldsymbol{A}\boldsymbol{x}
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% + \boldsymbol{u} \right) \\
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% \boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{A} \boldsymbol{x} - \boldsymbol{z}
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%.\end{align*}
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In order to express \ac{LP} decoding using \ac{ADMM} through proximal operators,
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it can be rewritten to fit the template for \textit{linearized \ac{ADMM}} given
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in \cite[Sec. 4.4.2]{proximal_algorithms}.
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We start with the general formulation of the \ac{LP} decoding problem:%
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%
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\begin{align*}
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\text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}} \\
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\text{subject to}\hspace{2mm} & \boldsymbol{T}_j \tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}
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\hspace{5mm} \forall j \in \mathcal{J}
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.\end{align*}
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%
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The constraints can be moved into the objective function: %
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%
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\begin{align}
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\begin{aligned}
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\text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
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+ \sum_{j\in \mathcal{J}} I_{P_{d_j}}\left(
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\boldsymbol{T}_j\tilde{\boldsymbol{c}} \right) \\
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\text{subject to}\hspace{2mm} & \tilde{\boldsymbol{c}} \in \mathbb{R}^n,
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\end{aligned}
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\label{eq:app:sum_reformulated}
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\end{align}%
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%
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using the \textit{indicator functions}
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$I_{\mathcal{P}_{d_j}} : \mathbb{R}^{d_j} \rightarrow \left\{ 0, +\infty \right\},
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\hspace{3mm} j\in \mathcal{J}$, defined as%
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%
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\begin{align*}
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I_{\mathcal{P}_{d_j}}\left( \boldsymbol{t} \right) :=
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\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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Further defining
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%
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\begin{align*}
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\boldsymbol{T} := \begin{bmatrix}
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\boldsymbol{T}_1 \\
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\boldsymbol{T}_2 \\
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\vdots \\
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\boldsymbol{T}_m
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\end{bmatrix}
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\hspace{5mm}\text{and}\hspace{5mm}
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g\left( \boldsymbol{t} \right) = \sum_{j\in\mathcal{J}} I_{\mathcal{P}_{d_j}}\left(
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\boldsymbol{B}_j \boldsymbol{t} \right)
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,\end{align*}%
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%
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\todo{Define $\boldsymbol{B}_j$}%
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problem (\ref{eq:app:sum_reformulated}) becomes%
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%
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\begin{align}
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\begin{aligned}
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\text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
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+ g\left( \boldsymbol{T}\tilde{\boldsymbol{c}} \right) \\
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\text{subject to}\hspace{2mm} & \tilde{\boldsymbol{c}} \in \mathbb{R}^n.
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\end{aligned}
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\label{eq:app:func_reformulated}
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\end{align}
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%
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\todo{Fix $\mu f$ and $\lambda g$ in the steps below}%
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In this form, it fits the template for linearized \ac{ADMM}.
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The iterative algorithm can then be expressed as%
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%
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\begin{align*}
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\tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
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- \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
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- \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
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\boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left(\boldsymbol{T}\tilde{\boldsymbol{c}}
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+ \boldsymbol{u} \right) \\
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\boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{T} \tilde{\boldsymbol{c}}
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- \boldsymbol{z}
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.\end{align*}
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%
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Using the definition of the proximal operator, the $\tilde{\boldsymbol{c}}$ update step
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can be rewritten to match the definition given in section \ref{sec:dec:LP Decoding using ADMM}:%
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%
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\begin{align*}
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\tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
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- \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
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- \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
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&= \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
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- \frac{\mu}{2} \left\Vert \boldsymbol{T}^\text{T}
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\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
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- \boldsymbol{z} + \boldsymbol{u} \right) \right\Vert_2^2 \right) \\
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&\overset{\text{(a)}}{=} \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}
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\tilde{\boldsymbol{c}}
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- \frac{\mu}{2} \left\Vert \boldsymbol{T}\tilde{\boldsymbol{c}}
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- \boldsymbol{z} + \boldsymbol{u} \right\Vert_2^2 \right) \\
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&= \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
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- \frac{\mu}{2} \left\Vert \begin{bmatrix}
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\boldsymbol{T}_1 \\
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\boldsymbol{T}_2 \\
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\vdots \\
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\boldsymbol{T}_m
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\end{bmatrix}
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\tilde{\boldsymbol{c}}
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- \begin{bmatrix}
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\boldsymbol{z}_1 \\
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\boldsymbol{z}_2 \\
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\vdots \\
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\boldsymbol{z}_m
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\end{bmatrix}
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+ \begin{bmatrix}
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\boldsymbol{u}_1 \\
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\boldsymbol{u}_2 \\
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\vdots \\
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\boldsymbol{u}_m
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\end{bmatrix} \right\Vert_2^2 \right)
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\hspace{5mm}\boldsymbol{z}_j,\boldsymbol{u}_j \in \mathbb{F}_2^{d_j},
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\hspace{2mm} j\in\mathcal{J}\\
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&= \argmin_{\tilde{\boldsymbol{c}}} \left( \boldsymbol{\gamma}^\text{T} \tilde{\boldsymbol{c}}
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- \frac{\mu}{2} \sum_{j \in J} \left\Vert \boldsymbol{T}_j \tilde{\boldsymbol{c}}
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- \boldsymbol{z}_j + \boldsymbol{u}_j \right\Vert_2^2 \right)
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.\end{align*}
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%
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Step (a) can be justified by observing that multiplication with $\boldsymbol{T}^\text{T}$
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only reorders components, leaving their values unchanged.
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Similarly to the $\boldsymbol{c}$ update, the $\boldsymbol{z}$ update step can be rewritten.
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Since $g\left( \cdot \right)$ is separable, so is its proximal operator
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\cite[Sec. 2.1]{proximal_algorithms}. The $\boldsymbol{z}$ update step can then
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be expressed as a number of smaller steps:%
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%
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\begin{gather*}
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\boldsymbol{z} \leftarrow \textbf{prox}_{\lambda g} \left(\boldsymbol{T}\tilde{\boldsymbol{c}}
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+ \boldsymbol{u} \right) \\[0.5em]
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\iff \\[0.5em]
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\begin{alignedat}{3}
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\boldsymbol{z}_j &\leftarrow \textbf{prox}_{\lambda I_{\mathcal{P}_{d_j}}}\left(
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\boldsymbol{T}_j \tilde{\boldsymbol{c}} + \boldsymbol{u}_j \right),
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\hspace{5mm} && \forall j\in\mathcal{J} \\
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& \overset{\text{(b)}}{=} \Pi_{\mathcal{P}_{d_j}}\left( \boldsymbol{T}_j
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\tilde{\boldsymbol{c}}
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+ \boldsymbol{u}_j \right), \hspace{5mm} && \forall j\in\mathcal{J}
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,\end{alignedat}
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\end{gather*}
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%
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where (b) results from the fact that appying the proximal operator on the
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indicator function of a convex set amounts to a projection onto the set
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\cite[Sec. 1.2]{proximal_algorithms}.
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@ -218,12 +218,8 @@
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\include{chapters/analysis_of_results}
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\include{chapters/analysis_of_results}
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\include{chapters/discussion}
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\include{chapters/discussion}
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\include{chapters/conclusion}
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\include{chapters/conclusion}
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\include{chapters/appendix}
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%
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% Appendix
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%
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\appendix
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%\listoffigures
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%\listoffigures
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%\listoftables
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%\listoftables
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