Diffusion Mean Estimation on the Diagonal of Product Manifolds

TL;DR

This paper introduces a stochastic simulation method for estimating diffusion means on product manifolds, avoiding costly nested optimizations required by traditional Fréchet mean computations.

Contribution

It develops a theoretical framework and two simulation schemes for estimating diffusion means without nested optimization, applicable to geometric statistics and deep learning.

Findings

Efficient stochastic schemes for mean estimation on manifolds.

Demonstrated applicability on two different manifolds.

Avoids nested optimization in mean computation.

Abstract

Computing sample means on Riemannian manifolds is typically computationally costly as exemplified by computation of the Fr\'echet mean which often requires finding minimizing geodesics to each data point for each step of an iterative optimization scheme. When closed-form expressions for geodesics are not available, this leads to a nested optimization problem that is costly to solve. The implied computational cost impacts applications in both geometric statistics and in geometric deep learning. The weighted diffusion mean offers an alternative to the weighted Fr\'echet mean. We show how the diffusion mean and the weighted diffusion mean can be estimated with a stochastic simulation scheme that does not require nested optimization. We achieve this by conditioning a Brownian motion in a product manifold to hit the diagonal at a predetermined time. We develop the theoretical foundation for the sampling-based mean estimation, we develop two simulation schemes, and we demonstrate the applicability of the method with examples of sampled means on two manifolds.