$\DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\Exp}{Exp} \newcommand{\geo}[2]{\gamma_{\overset{\frown}{#1,#2}}} \newcommand{\geoS}{\gamma} \newcommand{\geoD}[2]{\gamma_} \newcommand{\geoL}[2]{\gamma(#2; #1)} \newcommand{\gradM}{\nabla_{\M}} \newcommand{\gradMComp}[1]{\nabla_{\M,#1}} \newcommand{\Grid}{\mathcal G} \DeclareMathOperator{\Log}{Log} \newcommand{\M}{\mathcal M} \newcommand{\N}{\mathcal N} \newcommand{\mat}[1]{\mathbf{#1}} \DeclareMathOperator{\prox}{prox} \newcommand{\PT}[3]{\mathrm{PT}_{#1\to#2}#3} \newcommand{\R}{\mathbb R} \newcommand{\SPD}[1]{\mathcal{P}(#1)} \DeclareMathOperator{\Tr}{Tr} \newcommand{\tT}{\mathrm{T}} \newcommand{\vect}[1]{\mathbf{#1}}$

# The proximal map of an absolute difference squared

Proximal map $\prox_{\lambda f}$ for the function $f\colon\M^2\to\R,\quad f(x,y) = d^2_{\M}(x,y)$. Following [1], let $\geo{x}{y}$ denote the geodesic starting in $\geo{x}{y}(0)=x$ and reaching $\geo{x}{y}(1)=y$ at time $1$. Then

If a data item $x$ or $y$ contains a NaN it is set to the minimizer of the distance, i.e. initialized to the other argument of $f$.

### Matlab Documentation

 1 2 3 4 5 6 7 8 9 10 11 12 13 % proxAbsoluteDifference(M,f1,f2,lambda) prox of d^2_M(f1,f2) in both args % with parameter lambda on an arbitrary manifold. Values containing NaN % are initialized to the other argument, which is the minimizer. % % INPUT % M : A manifold % f1,f2 : data columns % lambda : proxParameter % % OUTPUT % x1,x2 : resulting columns of the proximal map % --- % Manifold-valued Image Restoration Toolbox 1.0 ~ R. Bergmann, 2014-10-19