Contractive Diffusion Probabilistic Models

Diffusion probabilistic models (DPMs) have emerged as a promising technology in generative modeling. The success of DPMs relies on two ingredients: time reversal of Markov diffusion processes and score matching. Most existing work implicitly assumes that score matching is close to perfect, while this assumption is questionable. In view of possibly unguaranteed score matching, we propose a new criterion -- the contraction of backward sampling in the design of DPMs. This leads to a novel class of contractive DPMs (CDPMs), including contractive Ornstein-Uhlenbeck (OU) processes and contractive sub-variance preserving (sub-VP) stochastic differential equations (SDEs). The key insight is that the contraction in the backward process narrows score matching errors, as well as discretization error. Thus, the proposed CDPMs are robust to both sources of error. Our proposal is supported by theoretical results, and is corroborated by experiments. Notably, contractive sub-VP shows the best performance among all known SDE-based DPMs on the CIFAR-10 dataset.