An Approximation Theory Framework for Measure-Transport Sampling Algorithms
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913504822296576 |
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| author | Baptista, Ricardo Hosseini, Bamdad Kovachki, Nikola B. Marzouk, Youssef M. Sagiv, Amir |
| author_facet | Baptista, Ricardo Hosseini, Bamdad Kovachki, Nikola B. Marzouk, Youssef M. Sagiv, Amir |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_13965 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An Approximation Theory Framework for Measure-Transport Sampling Algorithms Baptista, Ricardo Hosseini, Bamdad Kovachki, Nikola B. Marzouk, Youssef M. Sagiv, Amir Numerical Analysis Statistics Theory 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. |
| title | An Approximation Theory Framework for Measure-Transport Sampling Algorithms |
| topic | Numerical Analysis Statistics Theory |
| url | https://arxiv.org/abs/2302.13965 |