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Made proximal operator bold

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Andreas Tsouchlos 2023-04-02 01:02:09 +02:00
parent 7aa6248d71
commit d8646a2741
2 changed files with 4 additions and 4 deletions
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2
latex/thesis/chapters/decoding_techniques.tex
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@ -1017,7 +1017,7 @@ $\gamma h\left( \tilde{\boldsymbol{x}} \right) $ has to be computed.
It is then immediately approximated with gradient-descent:% It is then immediately approximated with gradient-descent:%
% %
\begin{align*} \begin{align*}
\text{prox}_{\gamma h} \left( \tilde{\boldsymbol{x}} \right) &\equiv \textbf{prox}_{\gamma h} \left( \tilde{\boldsymbol{x}} \right) &\equiv
\argmin_{\boldsymbol{t} \in \mathbb{R}^n} \argmin_{\boldsymbol{t} \in \mathbb{R}^n}
\left( \gamma h\left( \boldsymbol{t} \right) + \left( \gamma h\left( \boldsymbol{t} \right) +
\frac{1}{2} \lVert \boldsymbol{t} - \tilde{\boldsymbol{x}} \rVert \right)\\ \frac{1}{2} \lVert \boldsymbol{t} - \tilde{\boldsymbol{x}} \rVert \right)\\

6
latex/thesis/chapters/theoretical_background.tex
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@ -274,12 +274,12 @@ desired \cite[Sec. 15.3]{ryan_lin_2009}.
\textit{Proximal algorithms} are algorithms for solving convex optimization \textit{Proximal algorithms} are algorithms for solving convex optimization
problems, that rely on the use of \textit{proximal operators}. problems, that rely on the use of \textit{proximal operators}.
The proximal operator $\text{prox}_f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ The proximal operator $\textbf{prox}_f : \mathbb{R}^n \rightarrow \mathbb{R}^n$
of a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by of a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by
\cite[Sec. 1.1]{proximal_algorithms}% \cite[Sec. 1.1]{proximal_algorithms}%
% %
\begin{align*} \begin{align*}
\text{prox}_{\lambda f}\left( \boldsymbol{v} \right) = \argmin_{\boldsymbol{x}} \left( \textbf{prox}_{\lambda f}\left( \boldsymbol{v} \right) = \argmin_{\boldsymbol{x}} \left(
f\left( \boldsymbol{x} \right) + \frac{1}{2\lambda}\lVert \boldsymbol{x} f\left( \boldsymbol{x} \right) + \frac{1}{2\lambda}\lVert \boldsymbol{x}
- \boldsymbol{v} \rVert_2^2 \right) - \boldsymbol{v} \rVert_2^2 \right)
.\end{align*} .\end{align*}
@ -300,7 +300,7 @@ and minimizing $g$ using the proximal operator
% %
\begin{align*} \begin{align*}
\boldsymbol{x} \leftarrow \boldsymbol{x} - \lambda \nabla f\left( \boldsymbol{x} \right) \\ \boldsymbol{x} \leftarrow \boldsymbol{x} - \lambda \nabla f\left( \boldsymbol{x} \right) \\
\boldsymbol{x} \leftarrow \text{prox}_{\lambda g} \left( \boldsymbol{x} \right) \boldsymbol{x} \leftarrow \textbf{prox}_{\lambda g} \left( \boldsymbol{x} \right)
,\end{align*} ,\end{align*}
% %
Since $g$ is minimized with the proximal operator and is thus not required Since $g$ is minimized with the proximal operator and is thus not required