Deep nonlinear sufficient dimension reduction
提出一种结合广义鞅差散度与深度神经网络的非线性充分降维方法，理论证明其无偏性，优化方案比传统特征分解更高效灵活，估计误差的快速率接近非参数回归的极小极大速率。
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.