Comparing and interpolating distributions on manifold

TL;DR

This paper introduces a novel method for comparing and interpolating probability distributions on Riemannian manifolds by using covariance fields and similarity invariant functions, enabling meaningful comparisons despite local parametrization differences.

Contribution

It proposes a new approach to compare and interpolate distributions on manifolds using covariance fields and invariant functions, overcoming local parametrization issues.

Findings

Defined distances between distributions on manifolds.

Developed criteria for consistent interpolation of distributions.

Validated methods with experiments on the unit 2-sphere.

Abstract

We are interested in comparing probability distributions defined on Riemannian manifold. The traditional approach to study a distribution relies on locating its mean point and finding the dispersion about that point. On a general manifold however, even if two distributions are sufficiently concentrated and have unique means, a comparison of their covariances is not possible due to the difference in local parametrizations. To circumvent the problem we associate a covariance field with each distribution and compare them at common points by applying a similarity invariant function on their representing matrices. In this way we are able to define distances between distributions. We also propose new approach for interpolating discrete distributions and derive some criteria that assure consistent results. Finally, we illustrate with some experimental results on the unit 2-sphere.