diff --git a/latex/thesis/chapters/decoding_techniques.tex b/latex/thesis/chapters/decoding_techniques.tex index bc773f1..142fff7 100644 --- a/latex/thesis/chapters/decoding_techniques.tex +++ b/latex/thesis/chapters/decoding_techniques.tex @@ -1017,7 +1017,7 @@ $\gamma h\left( \tilde{\boldsymbol{x}} \right) $ has to be computed. It is then immediately approximated with gradient-descent:% % \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} \left( \gamma h\left( \boldsymbol{t} \right) + \frac{1}{2} \lVert \boldsymbol{t} - \tilde{\boldsymbol{x}} \rVert \right)\\ diff --git a/latex/thesis/chapters/theoretical_background.tex b/latex/thesis/chapters/theoretical_background.tex index bf685df..8db32f1 100644 --- a/latex/thesis/chapters/theoretical_background.tex +++ b/latex/thesis/chapters/theoretical_background.tex @@ -274,12 +274,12 @@ desired \cite[Sec. 15.3]{ryan_lin_2009}. \textit{Proximal algorithms} are algorithms for solving convex optimization 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 \cite[Sec. 1.1]{proximal_algorithms}% % \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} - \boldsymbol{v} \rVert_2^2 \right) .\end{align*} @@ -300,7 +300,7 @@ and minimizing $g$ using the proximal operator % \begin{align*} \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*} % Since $g$ is minimized with the proximal operator and is thus not required