DEEP NONLINEAR SUFFICIENT DIMENSION REDUCTION

• 成果类型: Article
• 署名作者: Chen, YinFeng; Jiao, YuLing; Qiu, Rui; Hu, Zhou
• 署名单位: East China Normal University; Wuhan University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2390
• 发表日期: 2024
• 页码: 1201-1226
• 关键词: sliced inverse regression; formulation; dependence

摘要

Linear sufficient dimension reduction, as exemplified by sliced inverse regression, has seen substantial development in the past thirty years. However, with the advent of more complex scenarios, nonlinear dimension reduction has gained considerable interest recently. This paper introduces a novel method for nonlinear sufficient dimension reduction, utilizing the generalized martingale difference divergence measure in conjunction with deep neural networks. The optimal solution of the proposed objective function is shown to be unbiased at the general level of sigma-fields. And two optimization schemes, based on the fascinating deep neural networks, exhibit higher efficiency and flexibility compared to the classical eigendecomposition of linear operators. Moreover, we systematically investigate the slow rate and fast rate for the estimation error based on advanced U-process theory. Remarkably, the fast rate almost coincides with the minimax rate of nonparametric regression. The validity of our deep nonlinear sufficient dimension reduction methods is demonstrated through simulations and real data analysis.