Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps
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arXiv
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| Format: | Preprint |
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2024
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| author | Baptista, Ricardo Pooladian, Aram-Alexandre Brennan, Michael Marzouk, Youssef Niles-Weed, Jonathan |
| author_facet | Baptista, Ricardo Pooladian, Aram-Alexandre Brennan, Michael Marzouk, Youssef Niles-Weed, Jonathan |
| contents | Conditional simulation is a fundamental task in statistical modeling: Generate samples from the conditionals given finitely many data points from a joint distribution. One promising approach is to construct conditional Brenier maps, where the components of the map pushforward a reference distribution to conditionals of the target. While many estimators exist, few, if any, come with statistical or algorithmic guarantees. To this end, we propose a non-parametric estimator for conditional Brenier maps based on the computational scalability of \emph{entropic} optimal transport. Our estimator leverages a result of Carlier et al. (2010), which shows that optimal transport maps under a rescaled quadratic cost asymptotically converge to conditional Brenier maps; our estimator is precisely the entropic analogues of these converging maps. We provide heuristic justifications for choosing the scaling parameter in the cost as a function of the number of samples by fully characterizing the Gaussian setting. We conclude by comparing the performance of the estimator to other machine learning and non-parametric approaches on benchmark datasets and Bayesian inference problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07154 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps Baptista, Ricardo Pooladian, Aram-Alexandre Brennan, Michael Marzouk, Youssef Niles-Weed, Jonathan Machine Learning Optimization and Control Conditional simulation is a fundamental task in statistical modeling: Generate samples from the conditionals given finitely many data points from a joint distribution. One promising approach is to construct conditional Brenier maps, where the components of the map pushforward a reference distribution to conditionals of the target. While many estimators exist, few, if any, come with statistical or algorithmic guarantees. To this end, we propose a non-parametric estimator for conditional Brenier maps based on the computational scalability of \emph{entropic} optimal transport. Our estimator leverages a result of Carlier et al. (2010), which shows that optimal transport maps under a rescaled quadratic cost asymptotically converge to conditional Brenier maps; our estimator is precisely the entropic analogues of these converging maps. We provide heuristic justifications for choosing the scaling parameter in the cost as a function of the number of samples by fully characterizing the Gaussian setting. We conclude by comparing the performance of the estimator to other machine learning and non-parametric approaches on benchmark datasets and Bayesian inference problems. |
| title | Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2411.07154 |