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Minor changes in appendix

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Andreas Tsouchlos 2023-04-07 11:17:50 +02:00
parent 0ba2120a21
commit a0a13dbb2d
1 changed files with 15 additions and 12 deletions
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27
latex/thesis/chapters/appendix.tex
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@ -89,15 +89,18 @@ problem (\ref{eq:app:sum_reformulated}) becomes%
In this form, it fits the template for linearized \ac{ADMM}.
The iterative algorithm can then be expressed as%
%
\begin{align*}
\tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
- \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
- \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
\boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left(\boldsymbol{T}\tilde{\boldsymbol{c}}
+ \boldsymbol{u} \right) \\
\boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{T} \tilde{\boldsymbol{c}}
- \boldsymbol{z}
.\end{align*}
\begin{align}
\begin{aligned}
\tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
- \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
- \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
\boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left(\boldsymbol{T}\tilde{\boldsymbol{c}}
+ \boldsymbol{u} \right) \\
\boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{T} \tilde{\boldsymbol{c}}
- \boldsymbol{z}.
\end{aligned}
\label{eq:app:admm_prox}
\end{align}
%
Using the definition of the proximal operator, the $\tilde{\boldsymbol{c}}$ update step
@ -134,12 +137,12 @@ can be rewritten to match the definition given in section \ref{sec:dec:LP Decodi
\boldsymbol{u}_2 \\
\vdots \\
\boldsymbol{u}_m
\end{bmatrix} \right\Vert_2^2 \right)
\end{bmatrix} \right\Vert_2^2 \right),
\hspace{5mm}\boldsymbol{z}_j,\boldsymbol{u}_j \in \mathbb{F}_2^{d_j},
\hspace{2mm} j\in\mathcal{J}\\
&= \argmin_{\tilde{\boldsymbol{c}}} \left( \boldsymbol{\gamma}^\text{T} \tilde{\boldsymbol{c}}
- \frac{\mu}{2} \sum_{j \in J} \left\Vert \boldsymbol{T}_j \tilde{\boldsymbol{c}}
- \boldsymbol{z}_j + \boldsymbol{u}_j \right\Vert_2^2 \right)
- \frac{\mu}{2} \sum_{j \in J} \left\Vert \boldsymbol{T}_j \tilde{\boldsymbol{c}}
- \boldsymbol{z}_j + \boldsymbol{u}_j \right\Vert_2^2 \right)
.\end{align*}
%
Step (a) can be justified by observing that multiplication with $\boldsymbol{T}^\text{T}$