STABILITY AND STATISTICAL INFERENCE FOR SEMIDISCRETE OPTIMAL TRANSPORT MAPS

• 成果类型: Article
• 署名作者: Sadhu, Ritwik; Goldfeld, Ziv; Kato, Kengo
• 署名单位: Cornell University; Cornell University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2104
• 发表日期: 2024
• 页码: 5694-5736
• 关键词: limit-theorems; CONVERGENCE; quantiles; bootstrap; algorithm; distance; ranks; cost

摘要

We study statistical inference for the optimal transport (OT) map (also known as the Brenier map) from a known absolutely continuous reference distribution onto an unknown finitely discrete target distribution. We derive limit distributions for the Lp-error with arbitrary p E [1, oc) and for linear functionals of the empirical OT map, together with their moment convergence. The former has a non-Gaussian limit, whose explicit density is derived, while the latter attains asymptotic normality. For both cases, we also establish consistency of the nonparametric bootstrap. The derivation of our limit theorems relies on new stability estimates of functionals of the OT map with respect to the dual potential vector, which may be of independent interest. We also discuss applications of our limit theorems to the construction of confidence sets for the OT map and inference for a maximum tail correlation. Finally, we show that, while the empirical OT map does not possess nontrivial weak limits in the L2 space, it satisfies a central limit theorem in a dual H & ouml;lder space, and the Gaussian limit law attains the asymptotic efficiency bound.