Operator-informed score matching for Markov diffusion models

Diffusion models are typically trained using score matching, yet score matching is agnostic to the particular forward process that defines the model. This paper argues that Markov diffusion models enjoy an advantage over other types of diffusion model, as their associated operators can be exploited to improve the training process. In particular, (i) there exists an explicit formal solution to the forward process as a sequence of time-dependent kernel mean embeddings; and (ii) the derivation of score-matching and related estimators can be streamlined. Building upon (i), we propose Riemannian diffusion kernel smoothing, which ameliorates the need for neural score approximation, at least in the low-dimensional context; Building upon (ii), we propose operator-informed score matching, a variance reduction technique that is straightforward to implement in both low- and high-dimensional diffusion modeling and is demonstrated to improve score matching in an empirical proof-of-concept.