Work

Nonasymptotic bounds for forward processes in denoising diffusions: Ornstein–Uhlenbeck is hard to beat

Year: 2025

Type: article

Abstract: Denoising diffusion probabilistic models (DDPMs) represent a recent advance in generative modelling that has delivered state-of-the-art results across many domains of applications. Despite their success, a rigorous theoretical understanding of the error within DDPMs, particularly the nonasymptotic bounds required for the comparison of their efficiency, remain scarce. Making minimal assumptions on the initial data distribution, allowing, for example, the manifold hypothesis, this paper presents explicit nonasymptotic bounds on the forward diffusion error in total variation (TV), expressed as a function of the terminal time T. We parametrise multi-modal data distributions in terms of the distance R to their furthest modes and consider forward diffusions with additive and multiplicative noise. Our analysis rigorously proves that, under mild assumptions, the canonical choice of the Ornstein–Uhlenbeck (OU) process cannot be significantly improved in terms of reducing the terminal time T as a function of R and error tolerance ε>0. Motivated by data distributions arising in generative modelling, we also establish a cut-off like phenomenon (as R→∞) for the convergence to its invariant measure in TV of an OU process, initialized at a multi-modal distribution with maximal mode distance R.

Source: The Annals of Applied Probability

DOI: 10.1214/25-aap2232

Authors: Miha Brešar, Aleksandar Mijatović

Institution: University of Warwick

Language: English

Topic: Markov Chains and Monte Carlo Methods

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Alan Turing Institute, Engineering and Physical Sciences Research Council, Division of Mathematical Sciences