Keywords: normalizing flow, score matching, Stein discrepancy
TL;DR: We theoretically build a connection between diffusion score matching with normalizing flows, and further extend it to continuous flows.
Abstract: Scoring matching (SM), and its related counterpart, Stein discrepancy (SD) have achieved great success in model training and evaluations. However, recent research shows their limitations when dealing with certain types of distributions. One possible fix is incorporating the original score matching (or Stein discrepancy) with a diffusion matrix, which is called diffusion score matching (DSM) (or diffusion Stein discrepancy (DSD)) . However, the lack of the interpretation of the diffusion limits its usage within simple distributions and manually chosen matrix. In this work, we plan to fill this gap by interpreting the diffusion matrix using normalizing flows. Specifically, we theoretically prove that DSM (or DSD) is equivalent to the original score matching (or score matching) evaluated in the transformed space defined by the normalizing flow, where the diffusion matrix is the inverse of the flow's Jacobian matrix. In addition, we also build its connection to Riemannian manifolds, and further extend it to continuous flows, where the change of DSM is characterized by an ODE.