
# 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 $\M$ 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 $\M$ 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 $\M$ for given input data $x\colon \Grid \to \M$, where $\Grid$ denotes the data domain —usually a signal or pixel grid, but necessarily on some array form—, a set $f_i$ of proximal maps, and a stopping criterion.

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The gradient descent algorithm on a manifold

Based on a manifold $\M$, a starting point $x^{(0)} = x$, and a gradient (or descent) direction $\gradM F$ of a function $F\colon\M\to\R$ this function performs a gradient descent algorithm

with step size $s_k$, $k=1,\ldots,$ until a stopping criterion is fulfilled.

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The subgradient descent algorithm on a manifold

Based on a manifold $\M$, a starting point $x^{(0)} = x$, and an element from the (sub)gradient (or descent) direction $\eta\in\partial F$ of a function $F\colon\M\to\R$ this function performs a subgradient descent algorithm %}

with step sizes $s_k$, $k=1,\ldots,$ where we keep track of the minimal value $x^{\mathrm{opt}}$. If $F$ is not single valued, this algorithm assumes, that it is meant as a parallel subgradient algorithm with respect to the last dimension(s) of $x$.

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