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

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.