OPTIMAL TRANSPORT MAP ESTIMATION IN GENERAL FUNCTION SPACES

• 成果类型: Article
• 署名作者: Divol, Vincent; Niles-Weed, Jonathan; Pooladian, Aram-Alexandre
• 署名单位: Institut Polytechnique de Paris; ENSAE Paris; New York University; New York University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2482
• 发表日期: 2025
• 页码: 963-988
• 关键词: INEQUALITIES; aggregation

摘要

We study the problem of estimating a function T, given independent samples from a distribution P and from the pushforward distribution TAP. This setting is motivated by applications in the sciences, where T represents the evolution of a physical system over time, and in machine learning, where, for example, T may represent a transformation learned by a deep neural network trained for a generative modeling task. To ensure identifiability, we assume that T = del phi 0 is the gradient of a convex function in which case T is known as an optimal transport map. Prior work has studied the estimation of T under the assumption that it lies in a H & ouml;lder class, but general theory is lacking. We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure P satisfy a Poincar & eacute; inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for H & ouml;lder transport maps but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when P is the normal distribution and the transport map is given by an infinite-width shallow neural network.