On the Representation and Learning of Monotone Triangular Transport Maps
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10208_2023_9630_ReferencePDF.pdf
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3.03 MB
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8017e3128dc4e36cd4d9eae15d240a8e
Author(s)
Baptista, Ricardo
Marzouk, Youssef
Zahm, Olivier
Date Issued
November 16, 2023
Journal
Foundations of Computational Mathematics
Publisher
Springer US
Citation
Baptista, R., Marzouk, Y. & Zahm, O. On the Representation and Learning of Monotone Triangular Transport Maps. Found Comput Math 24, 2063–2108 (2024).
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
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Creative Commons Attribution-Noncommercial-ShareAlike
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DOI of Published Version
https://doi.org/10.1007/s10208-023-09630-x
