Covariance fields

TL;DR

This paper introduces covariance fields on Riemannian manifolds, defining a tensor-based distribution characterization, and explores their properties, continuity, and the inverse problem of recovering distributions, highlighting differences from Euclidean spaces.

Contribution

It presents the concept of covariance fields on Riemannian manifolds and demonstrates their role as distribution representations, including solving the inverse problem.

Findings

Covariance fields are generally continuous on Riemannian manifolds.

Inverse problem of recovering distributions from covariance fields is solvable on non-Euclidean manifolds.

Covariance fields do not serve as distribution representations in Euclidean spaces.

Abstract

We introduce and study covariance fields of distributions on a Riemannian manifold. At each point on the manifold, covariance is defined to be a symmetric and positive definite (2,0)-tensor. Its product with the metric tensor specifies a linear operator on the respected tangent space. Collectively, these operators form a covariance operator field. We show that, in most circumstances, covariance fields are continuous. We also solve the inverse problem: recovering distribution from a covariance field. Surprisingly, this is not possible on Euclidean spaces. On non-Euclidean manifolds however, covariance fields are true distribution representations.