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：本文提出了一个通用的逼近论框架，用于分析概率建模中的测度传输算法。此类算法的一个主要动机应用是采样——统计推断与生成建模中的核心任务。我们给出了在连续极限下的先验误差估计，即当测度（或其密度）已知，但传输映射通过有限维函数空间离散化或近似时的情况。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论。分析中的第三个要素（具有独立研究价值）是发展了新的稳定性估计，将两个映射之间的距离与它们定义的推前测度之间的距离（或散度）联系起来。我们展示了该框架的一系列应用，通过Wasserstein度量、最大均值差异和Kullback–Leibler散度，为实际问题获得了定量收敛率。此外，我们还得到了流行的三角Knöthe-Rosenblatt映射逼近的特化率，并通过数值实验验证和扩展了我们的理论。