Infinite-dimensional generative diffusions via Doob's h-transform
This provides a rigorous and flexible approach for generative modeling in infinite-dimensional settings, addressing a limitation in existing diffusion methods.
The paper tackles the problem of defining generative diffusion models in infinite dimensions by introducing a framework using Doob's h-transform, which forces a reference diffusion towards a target distribution via an exponential change of measure, and validates it on synthetic and real data with established bounds.
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.