Optimal transport map estimation in general function spaces
研究了在一般函数空间中估计最优传输映射的统计方法，给出了比现有文献更弱的假设下的估计速率，适用于物理系统演化或生成模型等场景。
We study the problem of estimating a function T, given independent samples from a distribution P and from the pushforward distribution T♯P. 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=∇φ0 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 Hölder 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 Poincaré 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 Hölder 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.