2023/02/27 by Ricardo Baptista, Bamdad Hosseini, Baptista, Ricardo +7 · 1 citation
Computer Science · Decision Sciences · #FOS: Mathematics #Generative Adversarial Networks and Image Synthesis #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2302.13965
openalex publication_date 2023/02/27 · openalex created_date 2023/03/03 · openalex updated_date 2026/08/01
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 Knöthe-Rosenblatt maps are obtained, followed by numerical experiments that demonstrate and extend our theory.