Comparing and interpolating distributions on manifold

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.

💡 Research Summary

The paper tackles the fundamental problem of comparing and interpolating probability distributions that live on a Riemannian manifold, where the usual Euclidean tools based on a single mean and covariance matrix break down because each point has its own tangent space and local coordinate system. The authors introduce the notion of a covariance field: for a distribution p on a manifold M, a symmetric positive‑definite matrix Σₚ(x) is assigned to every point x∈M, representing how the mass of p spreads in the tangent space TₓM around x. Because all these matrices live in the same type of space (the space of symmetric tensors on a given tangent space), they can be compared pointwise if a suitable similarity‑invariant function I(·,·) is chosen. I must satisfy I(RARᵀ,RBRᵀ)=I(A,B) for any orthogonal transformation R, guaranteeing invariance under changes of local bases. The authors discuss several candidates (eigenvalue‑based L₂ distances, log‑determinant ratios, trace‑normalized metrics) and select a log‑determinant based measure that also respects the triangle inequality.

A global distance between two distributions p and q is then defined as the integral (or average) over the manifold of the pointwise invariant:
 D(p,q)=∫ₘ I(Σₚ(x), Σ_q(x)) dμ(x).
The paper proves that, under mild regularity conditions on I, D satisfies the axioms of a metric (non‑negativity, symmetry, triangle inequality). This provides a principled way to quantify how “far apart” two distributions are, without having to transport them to a common Euclidean chart.

Building on this distance, the authors propose a novel interpolation scheme for discrete distributions. Given two distributions p₁ and p₂ and an interpolation parameter t∈