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

# The proximal map of an absolute second order mixed difference

Proximal map $\prox_{\lambda f}$ for the function

Following  we can compute the gradient of $f$. Let $c=\geo{x}{z}(\tfrac{1}{2})$ and $\tilde c=\geo{y}{w}(\tfrac{1}{2})$ denote the mid points of geodesics, such that their distance $d_\M(c,\tilde c)$ is minimal among all geodesics connecting $x,z$ and $y,w$, respectively.

Then, using the adjoint Differential of the start point of a geodesic denoted by $J_{x,y}^*(t,\eta)$ the gradint is given by

The proximal map is computed approximately using a subgradient descent, where both the step size as well as the stopping criterion are optional parameters of this function.

### Matlab Documentation

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% proxAbsoluteSecondOrderMixedDifference(M,f1,f2,f3,f4,lambda)
% Compute the second order mixed difference based on the mid point model
% employing a sub gradient descent
%
% INPUT
%    M          : a manifold
%   f1,f2,f3,f4 : Data items from a 2x2 matrix each
%   lambda      : parameter of the proximal map
%
% OUTPUT
%  x1,x2,x3,x4 : result of the proximal map
% ---
% MVIRT | R. Bergmann, 2018-01-22