$ \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}} $

List of Algorithms

The following algorithms can be called with their parameters and optional values given in the usual Matlab way (a string, value alternating list). Then a parser is called and might perform additional validity checks. The second possibility is to defined a struct, e.g. with the field

1
  problem.M = Sn(2);

for data on the two-dimensional sphere. This does not evoke the parser and might include less validity checks. It should be preferred when performing a set of experiments.

The CPP algorithm for first and second order TV

This algorithm computes the cyclic or randomized proximal point algorithm on a manifold for the first and second order TV additive coupling functional.

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The CPP algorithm for Huber relaxed TV

This algorithm computes the cyclic or randomized proximal point algorithm on a manifold for the first order differences relaxed by a Huber functional.

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The Cyclic Proximal Point Algorithm

This algorithm computes the cyclic or randomized proximal point algorithm on a manifold for given input data , where denotes the data domain —usually a signal or pixel grid, but necessarily on some array form—, a set of proximal maps, and a stopping criterion.

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The gradient descent algorithm on a manifold

Based on a manifold , a starting point , and a gradient (or descent) direction of a function this function performs a gradient descent algorithm

with step size , until a stopping criterion is fulfilled.

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The subgradient descent algorithm on a manifold

Based on a manifold , a starting point , and an element from the (sub)gradient (or descent) direction of a function this function performs a subgradient descent algorithm %}

with step sizes , where we keep track of the minimal value . If is not single valued, this algorithm assumes, that it is meant as a parallel subgradient algorithm with respect to the last dimension(s) of .

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