Deep Barycentric Regression for Optimal Transport Map Estimation and its Statistical Optimality

2026-09-06

AI Summary

本文提出BROT方法，通过两步计算未正则化的最优传输计划并用深度神经网络拟合目标，以解决最优传输映射估计中的统计最优性和实际可操作性之间的差距。

Abstract

The optimal transport (OT) map provides a geometric transformation for aligning probability distributions and has become a useful tool in machine learning. However, existing estimators of the OT map still exhibit a gap between sharp statistical guarantees and practical parametric estimation based on stable training objectives. Theoretical estimators achieve minimax optimal convergence rates, but they are typically nonparametric and can incur demanding implementation design or inference costs. Practical estimators are parametric and scalable, but their statistical guarantees remain underexplored, and their min-max, adversarial-like training objectives can be sensitive to optimization algorithms. We propose BROT (Barycentric Regression for OT), a simple two-step method that first computes the unregularized OT plan and then fits a deep neural network (DNN) to the induced barycentric targets by least-squares regression. Under standard regularity conditions, we prove that the DNN estimator of BROT attains the minimax convergence rate, when the ground-truth OT map is Lipschitz. Numerical studies on synthetic datasets and an image dataset show that BROT provides accurate map estimates, strong target distribution matching, and competitive transport costs, compared to existing estimation methods. Experiments on two downstream tasks, single-cell perturbation prediction and unsupervised domain adaptation, further suggest that the accurate estimation of BROT can translate into stronger task performance.

Problem

• Optimal Transport
• Statistical Guarantees
• Parametric Estimation
• Minimax Convergence Rate

Innovation

• Barycentric Regression
• Optimal Transport Map Estimation
• Deep Neural Network
• Minimax Convergence Rate
• Lipschitz