Bridging linearity-based and kernel-based sufficient dimension reduction

TL;DR

This paper establishes a theoretical connection between linear and nonlinear sufficient dimension reduction methods using classical metric space and positive definite function results.

Contribution

It introduces a novel theoretical framework linking linear SDR techniques to nonlinear extensions through classical mathematical results.

Findings

Provides a unified view of linear and nonlinear SDR methods.

Highlights the role of metric spaces and positive definite functions in SDR.

Facilitates development of new SDR algorithms based on this connection.

Abstract

There has been a lot of interest in sufficient dimension reduction (SDR) methodologies as well as nonlinear extensions in the statistics literature. In this note, we use classical results regarding metric spaces and positive definite functions to link linear SDR procedures to their nonlinear counterparts.