Infinite-dimensional generative diffusions via Doob's h-transform

TL;DR

This paper develops a rigorous framework for infinite-dimensional generative diffusion models using Doob's h-transform, enabling greater flexibility and theoretical guarantees in high-dimensional settings.

Contribution

It introduces a novel infinite-dimensional diffusion construction via Doob's h-transform, extending existing methods and providing rigorous theoretical foundations.

Findings

Framework applicable to infinite-dimensional spaces

Method validated on synthetic and real data

Theoretical bounds established for the approach

Abstract

This paper introduces a rigorous framework for defining generative diffusion models in infinite dimensions via Doob's h-transform. Rather than relying on time reversal of a noising process, a reference diffusion is forced towards the target distribution by an exponential change of measure. Compared to existing methodology, this approach readily generalises to the infinite-dimensional setting, hence offering greater flexibility in the diffusion model. The construction is derived rigorously under verifiable conditions, and bounds with respect to the target measure are established. We show that the forced process under the changed measure can be approximated by minimising a score-matching objective and validate our method on both synthetic and real data.