论文标题
基于光滑的Wasserstein距离的生成建模的渐近保证
Asymptotic Guarantees for Generative Modeling Based on the Smooth Wasserstein Distance
论文作者
论文摘要
最小距离估计（MDE）作为（隐式）生成建模的表述获得了最近的关注。它认为，超过模型参数最小化经验数据分布与模型之间的统计距离。这种表述非常适合理论分析，但典型的结果受到维度的诅咒的阻碍。为了克服这一点并设计一个可扩展的有限样本统计MDE理论，我们采用了平滑1-Wasserstein距离（SWD）$ \ MATHSF {W} _1 _1^{（σ）} $的框架。最近显示，SWD可以保留经典瓦斯坦距离的度量和拓扑结构，同时享有无维度的经验收敛速度。在这项工作中，我们对最小光滑剂量估计量（MSWES）进行了彻底的统计研究，首先证明了估计量的可测量性和渐近一致性。然后，我们表征最佳模型参数的极限分布及其相关的最小SWD。这些结果暗示了基于MSWE的生成建模限制的$ O（N^{ - 1/2}）$概括，该建模符合任意维度。我们的主要技术工具是经验$ \ mathsf {w} _1^{（σ）} $的新颖高维极限分布结果。非等级极限的表征与经典的经验1-wasserstein距离形成鲜明对比，仅在一维情况下才知道相似的结果。我们理论的有效性得到了经验结果的支持，这使SWD作为在高维度中学习和推断的有效工具。
Minimum distance estimation (MDE) gained recent attention as a formulation of (implicit) generative modeling. It considers minimizing, over model parameters, a statistical distance between the empirical data distribution and the model. This formulation lends itself well to theoretical analysis, but typical results are hindered by the curse of dimensionality. To overcome this and devise a scalable finite-sample statistical MDE theory, we adopt the framework of smooth 1-Wasserstein distance (SWD) $\mathsf{W}_1^{(σ)}$. The SWD was recently shown to preserve the metric and topological structure of classic Wasserstein distances, while enjoying dimension-free empirical convergence rates. In this work, we conduct a thorough statistical study of the minimum smooth Wasserstein estimators (MSWEs), first proving the estimator's measurability and asymptotic consistency. We then characterize the limit distribution of the optimal model parameters and their associated minimal SWD. These results imply an $O(n^{-1/2})$ generalization bound for generative modeling based on MSWE, which holds in arbitrary dimension. Our main technical tool is a novel high-dimensional limit distribution result for empirical $\mathsf{W}_1^{(σ)}$. The characterization of a nondegenerate limit stands in sharp contrast with the classic empirical 1-Wasserstein distance, for which a similar result is known only in the one-dimensional case. The validity of our theory is supported by empirical results, posing the SWD as a potent tool for learning and inference in high dimensions.
