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

# Variance

Compute the variance of the data, i.e. the mean distance to the mean.

For given data $f\in\M^N$ let $\hat f$ denote the Riemannian center of mass. Then the variance is given by

Note that $n$ is the dimension of the manifold $\M$.

### Matlab Documentation

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% var(f) computes the empirical variance
%       1/(numel(f)-1) * sum (f-mean(f))^2
% of f
% INPUT
% f      : manifold valued Set
% OUTPUT
% v      : variance of the set (scalar)
% mean_f : mean value of the set
%
% ---
% Manifold-valued Image Restoration Toolbox 1.2
% J. Persch, R. bergmann, 2017-01-06