Covariance of centered distributions on manifold

TL;DR

This paper introduces a coordinate-independent family of distributions on Riemannian manifolds, relating their covariance to distribution concentration, with formulas for constant curvature spaces and validation through simulations.

Contribution

It proposes a coordinate-independent framework for distributions on manifolds using covariant tensors, enhancing versatility over existing coordinate-specific methods.

Findings

Derived relations between covariance and distribution concentration tensor.

Provided explicit formulas for constant curvature manifolds.

Validated theoretical results with simulations on sphere and hyperbolic plane.

Abstract

We define and study a family of distributions with domain complete Riemannian manifold. They are obtained by projection onto a fixed tangent space via the inverse exponential map. This construction is a popular choice in the literature for it makes it easy to generalize well known multivariate Euclidean distributions. However, most of the available solutions use coordinate specific definition that makes them less versatile. %We propose improvements in two directions. We define the distributions of interest in coordinate independent way by utilizing co-variant 2-tensors. Then we study the relation of these distributions to their Euclidean counterparts. In particular, we are interested in relating the covariance to the tensor that controls distribution concentration. We find approximating expression for this relation in general and give more precise formulas in case of manifolds of constant curvature, positive or negative. Results are confirmed by simulation studies of the standard normal distribution on the unit-sphere and hyperbolic plane.