A general theory for nonlinear sufficient dimension reduction: Formulation and estimation

TL;DR

This paper develops a comprehensive nonlinear sufficient dimension reduction framework, unifying and extending existing methods with new estimators that are computationally efficient and applicable to complex data structures.

Contribution

It introduces a general theory for nonlinear SDR, characterizes key classes, and proposes new estimators (GSIR and GSAVE) that are easy to compute and more effective.

Findings

GSIR accurately estimates the central class when complete and sufficient.

GSAVE captures larger portions of the central class when completeness fails.

The proposed methods outperform existing techniques in simulations and real data.

Abstract

In this paper we introduce a general theory for nonlinear sufficient dimension reduction, and explore its ramifications and scope. This theory subsumes recent work employing reproducing kernel Hilbert spaces, and reveals many parallels between linear and nonlinear sufficient dimension reduction. Using these parallels we analyze the properties of existing methods and develop new ones. We begin by characterizing dimension reduction at the general level of $\sigma$-fields and proceed to that of classes of functions, leading to the notions of sufficient, complete and central dimension reduction classes. We show that, when it exists, the complete and sufficient class coincides with the central class, and can be unbiasedly and exhaustively estimated by a generalized sliced inverse regression estimator (GSIR). When completeness does not hold, this estimator captures only part of the central class. However, in these cases we show that a generalized sliced average variance estimator (GSAVE) can capture a larger portion of the class. Both estimators require no numerical optimization because they can be computed by spectral decomposition of linear operators. Finally, we compare our estimators with existing methods by simulation and on actual data sets.