This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling -- a central task in statistical inference and generative modeling. We provide a priori error estimates in the continuum limit, i.e., when the measures (or their densities) are given, but when the transport map is discretized or approximated using a finite-dimensional function space. Our analysis relies on the regularity theory of transport maps and on classical approximation theory for high-dimensional functions. A third element of our analysis, which is of independent interest, is the development of new stability estimates that relate the distance between two maps to the distance~(or divergence) between the pushforward measures they define. We present a series of applications of our framework, where quantitative convergence rates are obtained for practical problems using Wasserstein metrics, maximum mean discrepancy, and Kullback--Leibler divergence. Specialized rates for approximations of the popular triangular Kn{\"o}the-Rosenblatt maps are obtained, followed by numerical experiments that demonstrate and extend our theory.

翻译：本文提出了一种通用的逼近理论框架，用于分析概率建模中的测度传输算法。此类算法的主要应用动机在于采样——这是统计推断与生成建模中的核心任务。我们在连续极限下提供了先验误差估计，即当测度（或其密度）给定，但传输映射通过有限维函数空间进行离散化或逼近时的理论分析。我们的分析依赖于传输映射的正则性理论以及高维函数的经典逼近理论。第三个分析要素是建立了新的稳定性估计，该估计将两个映射之间的距离与其定义的推前测度之间的距离（或散度）联系起来，这一结果本身具有独立的理论价值。我们展示了该框架的一系列应用实例，其中针对实际问题使用Wasserstein度量、最大均值差异和Kullback-Leibler散度获得了定量收敛速率。特别推导了流行的三角Knöthe-Rosenblatt映射逼近的专用收敛速率，并通过数值实验验证和拓展了我们的理论。