What's the score? Automated Denoising Score Matching for Nonlinear Diffusions

Reversing a diffusion process by learning its score forms the heart of diffusion-based generative modeling and for estimating properties of scientific systems. The diffusion processes that are tractable center on linear processes with a Gaussian stationary distribution. This limits the kinds of models that can be built to those that target a Gaussian prior or more generally limits the kinds of problems that can be generically solved to those that have conditionally linear score functions. In this work, we introduce a family of tractable denoising <PRE_TAG>score matching</POST_TAG> objectives, called local-DSM, built using local increments of the diffusion process. We show how local-DSM melded with Taylor expansions enables automated training and <PRE_TAG>score estimation</POST_TAG> with nonlinear diffusion processes. To demonstrate these ideas, we use automated-DSM to train generative models using non-<PRE_TAG>Gaussian priors</POST_TAG> on challenging low dimensional distributions and the CIFAR10 image dataset. Additionally, we use the automated-DSM to learn the scores for nonlinear processes studied in statistical physics.