Abstract

"Denoising diffusion probabilistic models (DDPMs) represent a recent advance in generative modelling that has delivered state-of-the-art results across many application domains. Despite their success, a rigorous theoretical understanding of the error within DDPMs, particularly the non-asymptotic 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 talk presents explicit non-asymptotic bounds on the forward diffusion error in total variation (TV), expressed as a function of the terminal time T.

The talk parametrises multi-modal data distributions in terms of the distance R to their furthest modes and consider forward diffusions with additive and multiplicative noise. The 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. Motivated by data distributions arising in generative modelling, the talk also establishes 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.

Joint work with M. Bresar."

About the speaker

Prof. Aleksandar Mijatović is a Professor of Probability at the Department of Statistics at the University of Warwick and a Fellow of The Alan Turing Institute in London. Prof. Mijatović was previously a Chair in Probability at the Department of Mathematics of King’s College London, and before that a Reader in Probability at the Mathematics Department of Imperial College London. Prof. Mijatović obtained his Ph.D. in low-dimensional topology at the University of Cambridge, before working in the City of London as a front-office quantitative analyst in Foreign Exchange derivative markets. His research interests are in Probability and its applications, including Stability of Stochastic Systems, Simulation and Monte Carlo Methods, Mathematical Finance, Numerical Stochastics, Data Science & Foundations of Machine Learning. He is also interested in the interactions of Probability with Analysis and Geometry.