A Riemannian covariance for manifold-valued data
Abstract
The extension of bivariate measures of dependence to non-Euclidean spaces is a challenging problem. The non-linear nature of these spaces makes the generalisation of classical measures of linear dependence (such as the covariance) not trivial. In this paper, we propose a novel approach to measure stochastic dependence between two random variables taking values in a Riemannian manifold, with the aim of both generalising the classical concepts of covariance and correlation and building a connection to Fr\'echet moments of random variables on manifolds. We introduce generalised local measures of covariance and correlation and we show that the latter is a natural extension of Pearson correlation. We then propose suitable estimators for these quantities and we prove strong consistency results. Finally, we demonstrate their effectiveness through simulated examples and a real-world application.
Cite
@article{arxiv.2410.06164,
title = {A Riemannian covariance for manifold-valued data},
author = {Meshal Abuqrais and Davide Pigoli},
journal= {arXiv preprint arXiv:2410.06164},
year = {2024}
}