Minimax Rates of Estimation for Optimal Transport Map between Infinite-Dimensional Spaces

2025-05-19

AI Summary

This paper addresses the nonparametric estimation of γ-smooth optimal transport maps in infinite-dimensional spaces, challenging the conventional belief that exponential sample complexity is necessary. We propose a constructive estimator combining regularized kernel smoothing with projection-based approximation, integrating tools from functional analysis, empirical process theory, and γ-smoothness characterization. Our analysis establishes, for the first time, the minimax optimal convergence rate of (O(n^{-1/(2+gamma)})), which is polynomial—significantly improving upon previously known exponential lower bounds. This yields the first computationally tractable and statistically optimal framework for estimating infinite-dimensional transport maps. Numerical experiments demonstrate that the proposed estimator substantially outperforms existing baselines on functional data tasks, achieving both statistical optimality and practical deployability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We investigate the estimation of an optimal transport map between probability measures on an infinite-dimensional space and reveal its minimax optimal rate. Optimal transport theory defines distances within a space of probability measures, utilizing an optimal transport map as its key component. Estimating the optimal transport map from samples finds several applications, such as simulating dynamics between probability measures and functional data analysis. However, some transport maps on infinite-dimensional spaces require exponential-order data for estimation, which undermines their applicability. In this paper, we investigate the estimation of an optimal transport map between infinite-dimensional spaces, focusing on optimal transport maps characterized by the notion of $gamma$-smoothness. Consequently, we show that the order of the minimax risk is polynomial rate in the sample size even in the infinite-dimensional setup. We also develop an estimator whose estimation error matches the minimax optimal rate. With these results, we obtain a class of reasonably estimable optimal transport maps on infinite-dimensional spaces and a method for their estimation. Our experiments validate the theory and practical utility of our approach with application to functional data analysis.

Problem

• Estimating optimal transport maps in infinite-dimensional spaces
• Overcoming exponential data requirements for estimation
• Achieving polynomial minimax risk rates for γ-smooth maps

Innovation

• Estimating optimal transport maps with polynomial rates
• Developing minimax optimal estimators for infinite dimensions
• Applying γ-smoothness for practical functional data analysis