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）在大量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在学习一族亚高斯概率分布时的逼近性与泛化性。我们引入了一种基于概率分布相对于标准高斯测度的相对密度的复杂度概念。我们证明：若对数相对密度能够被参数适当有界的神经网络局部逼近，则通过经验分数匹配生成的分布将以与维度无关的速率在总变差距离上逼近目标分布。我们通过包含某些高斯混合的例子阐释了该理论。证明的核心要素是推导出与前向过程相关的真实分数函数的无维度深度神经网络逼近率，这一结果本身也具有独立的研究价值。