Infinite-Dimensional Diffusion Models
This work addresses the problem of generating functions like images and time series for researchers and practitioners in machine learning, offering a novel approach to handle infinite-dimensional data more effectively.
The authors tackled the performance deterioration of diffusion models when applied to discretized infinite-dimensional data by directly formulating diffusion-based generative models in infinite dimensions, proving well-posedness and providing dimension-independent distance bounds. They developed design guidelines that align with canonical choices for images and improve upon them for other distributions, demonstrating results theoretically and empirically on manifold data and Bayesian inverse problems.
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.