Score matching for sub-Riemannian bridge sampling

TL;DR

This paper develops a novel score matching method for simulating diffusion bridges on sub-Riemannian manifolds, overcoming geometric and hypoelliptic challenges with machine learning techniques.

Contribution

It introduces a new approach to train score approximators on sub-Riemannian manifolds by generalizing denoising loss and applying stochastic Taylor expansion.

Findings

Successfully demonstrated on the Heisenberg group

Generated samples from the bridge process

Showed concentration of the process for small time

Abstract

Simulation of conditioned diffusion processes is an essential tool in inference for stochastic processes, data imputation, generative modelling, and geometric statistics. Whilst simulating diffusion bridge processes is already difficult on Euclidean spaces, when considering diffusion processes on Riemannian manifolds the geometry brings in further complications. In even higher generality, advancing from Riemannian to sub-Riemannian geometries introduces hypoellipticity, and the possibility of finding appropriate explicit approximations for the score of the diffusion process is removed. We handle these challenges and construct a method for bridge simulation on sub-Riemannian manifolds by demonstrating how recent progress in machine learning can be modified to allow for training of score approximators on sub-Riemannian manifolds. Since gradients dependent on the horizontal distribution, we generalise the usual notion of denoising loss to work with non-holonomic frames using a stochastic Taylor expansion, and we demonstrate the resulting scheme both explicitly on the Heisenberg group and more generally using adapted coordinates. We perform numerical experiments exemplifying samples from the bridge process on the Heisenberg group and the concentration of this process for small time.