一种可统一处理线性与非线性降维的新神经网络方法
Belted and Ensembled Neural Network for Linear and Nonlinear Sufficient Dimension Reduction
- 通过窄层（belt）与响应变换族（ensemble）构建神经网络框架
- 支持条件均值与条件分布的降维，计算速度远超传统方法
- 适合需要高效降维且目标灵活的数据分析场景
我们提出一种统一、灵活且易于实现的充分降维框架，可同时处理线性与非线性降维，以及条件分布与条件均值作为估计目标。该框架基于一种特殊结构的神经网络——束带集成神经网络（BENN），包含一个窄的潜在层（称为belt）和一组响应变换（称为ensemble）。通过在神经网络不同层放置belt，可实现线性或非线性充分降维；通过选择合适的变换族，可针对条件分布或条件均值进行降维。得益于神经网络的优势，该方法计算速度快，克服了传统充分降维估计器需对p或n维矩阵求逆的计算瓶颈。我们开发了算法并分析了收敛速率，与现有方法对比，并在两个数据例中应用。
原文摘要 · Abstract (English)
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.
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