$ \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 the distance

Proximal map for the function for some fixed value . Following [1], let denote the geodesic starting in and reaching at time . Then

If a data item contains a NaN it is set to the minimizer of the distance, i.e. initialized to the fixed item. Values can be kept unchanged by activating the optional FixedMask boolean array.

Matlab Documentation

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% proxDistanceSquared(M,x,f,lambda) prox of f(x) = d(x,y), y fixed,
% and prox paramater lambda. The fixedMask can be used to fix values, i.e.
% they are kept unchanged by hard reset.
%
% INPUT
%    M    : a manifold
%  x,f  : data point( sets/columns ) on the maifold M
%  lambda : stepsize towards f
%
% OPTIONAL
% FixedMask : fix a certain subset of x
%
% OUTPUT
%       x : result point( sets) of the proximal map
%       w : indicator which terms where proxed, here: all
% ---
% Manifold-valued Image Restoration Toolbox 1.0 ~ R. Bergmann, 2014-10-19

See also

References

  1. Ferreira, O P and Oliveira, P R (2002). Proximal point algorithm on Riemannian manifolds. Optimization. 51 257–70