Abstract
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 σ-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.
Funding Statement
Yinfeng Chen, Rui Qiu and Zhou Yu are supported by the National Key R&D Program of China (Grant No. 2023YFA1008700 and 2023YFA1008703), the National Natural Science Foundation of China (Grant No. 12371289), the Shanghai Pilot Program for Basic Research (Grant No. TQ20220105) and and the 111 project (B14019).
Yuling Jiao is supported by the National Natural Science Foundation of China (Grant No. 12371441), the Fundamental Research Funds for the Central Universities and the research fund of KLATASDSMOE of China.
Acknowledgments
The authors thank the Editor, Associate Editor, and three anonymous reviewers for their constructive feedback on earlier versions of this paper. The authors contributed equally to this paper and are listed in alphabetical order. All correspondence should be addressed to Zhou Yu (the corresponding author) at zyu@stat.ecnu.edu.cn.
Citation
YinFeng Chen. YuLing Jiao. Rui Qiu. Zhou Yu. "Deep nonlinear sufficient dimension reduction." Ann. Statist. 52 (3) 1201 - 1226, June 2024. https://doi.org/10.1214/24-AOS2390
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