We introduce a unified, flexible, and easy-to-implement framework of sufficient dimension reduction that can accommodate both linear and nonlinear dimension reduction, and both the conditional distribution and the conditional mean as the targets of estimation. This unified framework is achieved by a specially structured neural network -- the Belted and Ensembled Neural Network (BENN) -- that consists of a narrow latent layer, which we call the belt, and a family of transformations of the response, which we call the ensemble. By strategically placing the belt at different layers of the neural network, we can achieve linear or nonlinear sufficient dimension reduction, and by choosing the appropriate transformation families, we can achieve dimension reduction for the conditional distribution or the conditional mean. Moreover, thanks to the advantage of the neural network, the method is very fast to compute, overcoming a computation bottleneck of the traditional sufficient dimension reduction estimators, which involves the inversion of a matrix of dimension either p or n. We develop the algorithm and convergence rate of our method, compare it with existing sufficient dimension reduction methods, and apply it to two data examples.

翻译：本文提出了一种统一、灵活且易于实现的充分降维框架，该框架能够同时适应线性和非线性降维，并以条件分布和条件均值作为估计目标。这一统一框架通过一种特殊结构的神经网络——带式集成神经网络（BENN）——实现，该网络包含一个狭窄的潜在层（我们称之为"带"）以及响应变量的一系列变换（我们称之为"集成"）。通过策略性地将"带"置于神经网络的不同层，我们可以实现线性或非线性充分降维；通过选择适当的变换族，我们可以实现针对条件分布或条件均值的降维。此外，得益于神经网络的优势，该方法计算速度极快，克服了传统充分降维估计器涉及维度为 p 或 n 的矩阵求逆的计算瓶颈。我们发展了该方法的算法与收敛速率，将其与现有充分降维方法进行比较，并应用于两个数据实例。