  Diffusion models have recently outperformed alternative approaches to model
the distribution of natural images, such as GANs. 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 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 theoretically and by
extensive numerical experiments.
