TY - UNPB

T1 - An Approximation Theory Framework for Measure-Transport Sampling Algorithms

AU - Baptista, Ricardo

AU - Hosseini, Bamdad

AU - Kovachki, Nikola B.

AU - Marzouk, Youssef M.

AU - Sagiv, Amir

N1 - Pre-print from ArXiv: https://arxiv.org/abs/2302.13965

PY - 2023

Y1 - 2023

N2 - 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{ö}the-Rosenblatt maps are obtained, followed by numerical experiments that demonstrate and extend our theory.

AB - 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{ö}the-Rosenblatt maps are obtained, followed by numerical experiments that demonstrate and extend our theory.

U2 - 10.48550/arXiv.2302.13965

DO - 10.48550/arXiv.2302.13965

M3 - פרסום מוקדם

BT - An Approximation Theory Framework for Measure-Transport Sampling Algorithms

ER -