A Riemannian Covariance for Manifold-Valued Data

Abstract The extension of bivariate measures of dependence to non-Euclidean spaces is a challenging problem. The nonlinear nature of these spaces makes the generalisation of classical measures of linear dependence (such as the covariance) not trivial. Examples of non-Euclidean data in imaging are shapes, diffusion tensor imaging and directional images. 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 Fréchet 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.