{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/an-approximation-theory-framework-for-measure","title":"An Approximation Theory Framework for Measure-Transport Sampling Algorithms","arxiv_id":"2302.13965","date":"2023-02-27","proceeding":null,"authors":["Ricardo Baptista","Bamdad Hosseini","Nikola B. Kovachki","Youssef M. Marzouk","Amir Sagiv"],"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.","url_abs":"https://arxiv.org/abs/2302.13965v4","url_pdf":"https://arxiv.org/pdf/2302.13965v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"an-approximation-theory-framework-for-measure","repo_url":"https://github.com/baptistar/transportmapapproximation","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}