Score-based generative models break the curse of dimensionality in learning a family of sub-Gaussian probability distributions
This provides theoretical guarantees for SGMs in high-dimensional settings, addressing a gap in their mathematical understanding, though it is incremental as it builds on existing SGM frameworks.
The paper tackles the problem of establishing mathematical foundations for score-based generative models (SGMs) by analyzing their approximation and generalization in learning sub-Gaussian probability distributions, proving that under certain conditions, the generated distribution approximates the target with a dimension-independent rate in total variation.
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.