PC Seminar: Statistical Guarantees for Denoising Reflected Diffusion Models

Abstract: Diffusion-based generative models offer a highly flexible approach to modelling and generating complex data distributions, yet their theoretical properties on bounded domains remain only partially understood. We study a class of denoising reflected diffusion models on bounded domains, which address the practical limitations of conventional designs operating in unbounded state spaces. A primary mathematical challenge in this setting is the absence of Gaussian transition kernels. To overcome this, we employ infinite series expansions and spectral methods, combined with a rigorous analysis of sparse neural networks, to approximate the score function and control the resulting approximation error. For target distributions under Sobolev smoothness assumptions, we establish near-minimax optimal convergence rates in total variation and Wasserstein distances, demonstrating full adaptivity to the intrinsic dimension of the underlying subspace. These results confirm that incorporating reflecting boundaries preserves the statistical efficiency of the underlying diffusion processes, matching the convergence behaviour known for unconstrained settings.

Joint work with Asbjørn Holk Thomsen and Lukas Trottner.

This is a seminar in our seminar series on Probability and Combinatorics (PC).