Bayesian Quadrature on Riemannian Data Manifolds

TL;DR

This paper introduces a Bayesian quadrature method tailored for Riemannian manifolds, enabling efficient numerical integration of statistical models on complex geometric data structures, with applications in molecular dynamics.

Contribution

It develops a probabilistic numerical approach using Bayesian quadrature for integration on Riemannian manifolds, reducing computational costs compared to traditional methods.

Findings

Bayesian quadrature outperforms Monte Carlo methods in efficiency.

Active exploration significantly reduces the number of function evaluations.

Application to molecular dynamics demonstrates practical benefits.

Abstract

Riemannian manifolds provide a principled way to model nonlinear geometric structure inherent in data. A Riemannian metric on said manifolds determines geometry-aware shortest paths and provides the means to define statistical models accordingly. However, these operations are typically computationally demanding. To ease this computational burden, we advocate probabilistic numerical methods for Riemannian statistics. In particular, we focus on Bayesian quadrature (BQ) to numerically compute integrals over normal laws on Riemannian manifolds learned from data. In this task, each function evaluation relies on the solution of an expensive initial value problem. We show that by leveraging both prior knowledge and an active exploration scheme, BQ significantly reduces the number of required evaluations and thus outperforms Monte Carlo methods on a wide range of integration problems. As a concrete application, we highlight the merits of adopting Riemannian geometry with our proposed framework on a nonlinear dataset from molecular dynamics.