Infinite-Dimensional Diffusion Models

TL;DR

This paper develops and analyzes diffusion models directly in infinite-dimensional spaces, enabling better generative modeling of functions and complex data types without discretization issues.

Contribution

It introduces a well-posed infinite-dimensional diffusion framework with dimension-independent bounds and provides practical guidelines for designing such models.

Findings

The models are well-posed in infinite dimensions.

Dimension-independent distance bounds are established.

Empirical results on manifolds and inverse problems demonstrate effectiveness.

Abstract

Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply diffusion models to the discretized data. While such approaches are practically appealing, the performance of the resulting algorithms typically deteriorates as discretization parameters are refined. In this paper, we instead directly formulate diffusion-based generative models in infinite dimensions and apply them to the generative modelling of functions. We prove that our formulations are well posed in the infinite-dimensional setting and provide dimension-independent distance bounds from the sample to the target measure. Using our theory, we also develop guidelines for the design of infinite-dimensional diffusion models. For image distributions, these guidelines are in line with current canonical choices. For other distributions, however, we can improve upon these canonical choices. We demonstrate these results both theoretically and empirically, by applying the algorithms to data distributions on manifolds and to distributions arising in Bayesian inverse problems or simulation-based inference.