Score Matching Riemannian Diffusion Means

TL;DR

This paper introduces an efficient score matching approach for estimating Riemannian diffusion means, enabling faster computation of means and related operations on complex manifolds without closed-form distances.

Contribution

It presents a novel score matching method for Riemannian diffusion means that is more efficient than Monte Carlo methods and applicable to learned manifolds.

Findings

More efficient than Monte Carlo simulation.

Retains accuracy in estimating diffusion means.

Extends to computing Fréchet mean and logarithmic map.

Abstract

Estimating means on Riemannian manifolds is generally computationally expensive because the Riemannian distance function is not known in closed-form for most manifolds. To overcome this, we show that Riemannian diffusion means can be efficiently estimated using score matching with the gradient of Brownian motion transition densities using the same principle as in Riemannian diffusion models. Empirically, we show that this is more efficient than Monte Carlo simulation while retaining accuracy and is also applicable to learned manifolds. Our method, furthermore, extends to computing the Fr\'echet mean and the logarithmic map for general Riemannian manifolds. We illustrate the applicability of the estimation of diffusion mean by efficiently extending Euclidean algorithms to general Riemannian manifolds with a Riemannian $k$-means algorithm and maximum likelihood Riemannian regression.