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.

翻译：尽管基于分数的生成模型（SGMs）在众多图像生成任务中取得了显著成功，但其数学基础仍然有限。本文分析了SGMs在学习一类子高斯概率分布时的逼近与泛化能力。我们引入了一个关于概率分布相对于标准高斯测度的相对密度的复杂性概念。我们证明：若对数相对密度可通过参数适度有界的神经网络进行局部逼近，则通过经验分数匹配生成的分布能够以与维度无关的速率在总变差距离下逼近目标分布。我们通过包含特定高斯混合模型在内的实例验证了该理论。证明的关键要素在于推导前向过程中真实分数函数的无维度深度神经网络逼近率，这一结论本身具有独立的研究价值。