Understanding DDPM Latent Codes Through Optimal Transport
This work provides theoretical insights into diffusion models, which are important for practical applications like likelihood estimation, but it is incremental as it partially addresses an open question for a specific case.
The authors tackled the problem of understanding the theoretical properties of the encoder map in diffusion models, specifically for the VP SDE (DDPM) approach, and showed that it coincides with the optimal transport map for common distributions, supported by theoretical analysis and numerical experiments.
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.