  While score-based generative models (SGMs) have achieved remarkable success
in enormous image generation tasks, their mathematical foundations are still
limited. In this paper, we analyze the approximation and generalization of SGMs
in learning a family of sub-Gaussian probability distributions. We introduce a
notion of complexity for probability distributions in terms of their relative
density with respect to the standard Gaussian measure. We prove that if the
log-relative density can be locally approximated by a neural network whose
parameters can be suitably bounded, then the distribution generated by
empirical score matching approximates the target distribution in total
variation with a dimension-independent rate. We illustrate our theory through
examples, which include certain mixtures of Gaussians. An essential ingredient
of our proof is to derive a dimension-free deep neural network approximation
rate for the true score function associated with the forward process, which is
interesting in its own right.
