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

# The proximal map of an absolute difference with Huber relaxation

Proximal map $\prox_{\lambda f}$ for the function $f\colon\M^2\to\R,\quad f(x,y) = h(d_{\M}(x,y))$, where $h$ relaxes the nondifferential point at $d=0$. The function $h$ is given as in [1] by

For small $s$ we obtain a parabola and for large s a linear increase with ascent $\omega\tau\sqrt{2}$. From both the proximal map of an absolute difference and its squared. The closed form of the proximal map also results from these two as

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

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% proxAbsoluteDifferenceHuber(M,f1,f2,lambda,tau,omega) prox d(x,y) relaxed
% Given the Huber function
% h(t) = tau^2t^2 for t< omega/(sqrt(2)tau), omega*sqrt(2)*tau*t - omega^2/2
% i.e. for small t a suqared function (steered by tau) and for large t a
% linear part with ascent omega.
% lambda is the prox parameter, as for the absolute difference, this
% function inpaints NaN values and works on arbitrary manifolds
%
% INPUT
%   M         : A manifold
%  f1,f2   : data columns
%  lambda     : parameter within the proximal map
%  tau, omega : parameters of the Huber function
%
% OUTPUT
%  x1,x2   : resulting columns of the proximal map
% ---
% Manifold-valued Image Restoration Toolbox 1.0 ~ R. Bergmann, 2014-10-19