On the Representation and Learning of Monotone Triangular Transport Maps
Transportation of measure provides a versatile approach for modeling complex probability distributions, with applications in density estimation, Bayesian inference, generative modeling, and beyond. Monotone triangular transport maps—approximations of the Knothe–Rosenblatt (KR) rearrangement—are a ca...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer US
2024
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| Online Access: | https://hdl.handle.net/1721.1/157755 |
| _version_ | 1824458049833664512 |
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| author | Baptista, Ricardo Marzouk, Youssef Zahm, Olivier |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Baptista, Ricardo Marzouk, Youssef Zahm, Olivier |
| author_sort | Baptista, Ricardo |
| collection | MIT |
| description | Transportation of measure provides a versatile approach for modeling complex probability distributions, with applications in density estimation, Bayesian inference, generative modeling, and beyond. Monotone triangular transport maps—approximations of the Knothe–Rosenblatt (KR) rearrangement—are a canonical choice for these tasks. Yet the representation and parameterization of such maps have a significant impact on their generality and expressiveness, and on properties of the optimization problem that arises in learning a map from data (e.g., via maximum likelihood estimation). We present a general framework for representing monotone triangular maps via invertible transformations of smooth functions. We establish conditions on the transformation such that the associated infinite-dimensional minimization problem has no spurious local minima, i.e., all local minima are global minima; and we show for target distributions satisfying certain tail conditions that the unique global minimizer corresponds to the KR map. Given a sample from the target, we then propose an adaptive algorithm that estimates a sparse semi-parametric approximation of the underlying KR map. We demonstrate how this framework can be applied to joint and conditional density estimation, likelihood-free inference, and structure learning of directed graphical models, with stable generalization performance across a range of sample sizes. |
| first_indexed | 2025-02-19T04:19:43Z |
| format | Article |
| id | mit-1721.1/157755 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2025-02-19T04:19:43Z |
| publishDate | 2024 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1577552024-12-21T06:02:26Z On the Representation and Learning of Monotone Triangular Transport Maps Baptista, Ricardo Marzouk, Youssef Zahm, Olivier Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Transportation of measure provides a versatile approach for modeling complex probability distributions, with applications in density estimation, Bayesian inference, generative modeling, and beyond. Monotone triangular transport maps—approximations of the Knothe–Rosenblatt (KR) rearrangement—are a canonical choice for these tasks. Yet the representation and parameterization of such maps have a significant impact on their generality and expressiveness, and on properties of the optimization problem that arises in learning a map from data (e.g., via maximum likelihood estimation). We present a general framework for representing monotone triangular maps via invertible transformations of smooth functions. We establish conditions on the transformation such that the associated infinite-dimensional minimization problem has no spurious local minima, i.e., all local minima are global minima; and we show for target distributions satisfying certain tail conditions that the unique global minimizer corresponds to the KR map. Given a sample from the target, we then propose an adaptive algorithm that estimates a sparse semi-parametric approximation of the underlying KR map. We demonstrate how this framework can be applied to joint and conditional density estimation, likelihood-free inference, and structure learning of directed graphical models, with stable generalization performance across a range of sample sizes. 2024-12-05T15:37:43Z 2024-12-05T15:37:43Z 2023-11-16 2024-12-05T09:31:22Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/157755 Baptista, R., Marzouk, Y. & Zahm, O. On the Representation and Learning of Monotone Triangular Transport Maps. Found Comput Math 24, 2063–2108 (2024). en https://doi.org/10.1007/s10208-023-09630-x Foundations of Computational Mathematics Creative Commons Attribution-Noncommercial-ShareAlike http://creativecommons.org/licenses/by-nc-sa/4.0/ SFoCM application/pdf Springer US Springer US |
| spellingShingle | Baptista, Ricardo Marzouk, Youssef Zahm, Olivier On the Representation and Learning of Monotone Triangular Transport Maps |
| title | On the Representation and Learning of Monotone Triangular Transport Maps |
| title_full | On the Representation and Learning of Monotone Triangular Transport Maps |
| title_fullStr | On the Representation and Learning of Monotone Triangular Transport Maps |
| title_full_unstemmed | On the Representation and Learning of Monotone Triangular Transport Maps |
| title_short | On the Representation and Learning of Monotone Triangular Transport Maps |
| title_sort | on the representation and learning of monotone triangular transport maps |
| url | https://hdl.handle.net/1721.1/157755 |
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