Understanding DDPM Latent Codes Through Optimal Transport

Abstract

Diffusion models have recently outperformed alternative approaches to model the distribution of natural images. Such diffusion models allow for deterministic sampling via the probability flow ODE, giving rise to a latent space and an encoder map. While having important practical applications, such as the estimation of the likelihood, the theoretical properties of this map are not yet fully understood. In the present work, we partially address this question for the popular case of the VP-SDE (DDPM) approach. We show that, perhaps surprisingly, the DDPM encoder map coincides with the optimal transport map for common distributions; we support this claim by extensive numerical experiments using advanced tensor train solver for multidimensional Fokker-Planck equation. We provide additional theoretical evidence for the case of multivariate normal distributions.