$ \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 Tutorials & Examples

Tutorials

Tutorial illustration
Illustrating the gradient of a second order difference

This example illustrates how to compute a gradient of a second order difference and how to vizualize the result using an export to Asymptote.

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Tutorial illustration
Computing the mean with gradient descent

This tutorial introduces the Riemannian center of mass briefly and how to compute this generalization of a mean on an arbitrary manifold using a gradient descent algorithm.

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Tutorial illustration
Computing the median with cyclic proximal point

This tutorial introduces the cyclic proximal point algorithm using again the example of the Riemannian median that can be introduced analogously to the Riemannian center of mass.

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Examples

Example illustration
Bernoulli's Lemniscate on the Sphere

The example of Bernoulli’s Lemniscate on the sphere denoises a signal which is sampled from the curve defined as Bernoulli’s Lemniscate in the tangential space of the north pole and then brought onto the manifold by the exponential map. This signal is obstructed by noise and the example denoises this noisy artificial signal using the CPP algorithm for the additive first and second order differences model. The script itself performs a small parameter test.

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