Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps

• URL: http://arxiv.org/abs/2411.07154v1
• Date: Mon, 11 Nov 2024 17:32:47 GMT
• Title: Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps
• Authors: Ricardo Baptista, Aram-Alexandre Pooladian, Michael Brennan, Youssef Marzouk, Jonathan Niles-Weed,
• Abstract summary: Conditional simulation is a fundamental task in statistical modeling. One promising approach is to construct conditional Brenier maps, where the components of the map pushforward a reference distribution to conditionals of the target. We propose a non-parametric estimator for conditional Brenier maps based on the computational scalability of emphentropic optimal transport.
• Score: 13.355769319031184
• License:
• Abstract: 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.

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