A Riemannian covariance for manifold-valued data

TL;DR

This paper introduces a novel Riemannian covariance measure for manifold-valued data, extending classical dependence measures to non-Euclidean spaces with proven consistency and practical demonstrations.

Contribution

It proposes a new Riemannian covariance and correlation framework for manifold-valued data, connecting to Fréchet moments and providing estimators with strong consistency.

Findings

Effective in simulated examples

Applicable to real-world data

Generalizes classical dependence measures

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.