An Approximation Theory Framework for Measure-Transport Sampling Algorithms

TL;DR

This paper introduces an approximation-theoretic framework for analyzing measure-transport algorithms used in sampling, providing error estimates, stability results, and convergence rates applicable to various divergences and practical algorithms.

Contribution

It develops a general theoretical framework combining approximation theory and transport map regularity to analyze measure-transport sampling algorithms.

Findings

Provides a priori error estimates in the continuum limit.

Derives convergence rates for Wasserstein, MMD, and KL divergences.

Demonstrates the theory with numerical experiments on Knöthe-Rosenblatt maps.

Abstract

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.