Principal support vector machines for linear and nonlinear sufficient dimension reduction

TL;DR

This paper introduces Principal Support Vector Machines (PSVM), a novel method for linear and nonlinear sufficient dimension reduction that uses support vector machines and kernel methods to identify the reduction space with strong theoretical guarantees.

Contribution

The paper proposes a new PSVM approach for both linear and nonlinear dimension reduction, providing theoretical properties and demonstrating practical advantages over existing methods.

Findings

PSVM provides unbiased, $ extit{sqrt{n}}$-consistent, asymptotically normal estimators.

The method extends to nonlinear reduction using reproducing kernel Hilbert spaces.

Empirical results show PSVM outperforms existing dimension reduction techniques.

Abstract

We introduce a principal support vector machine (PSVM) approach that can be used for both linear and nonlinear sufficient dimension reduction. The basic idea is to divide the response variables into slices and use a modified form of support vector machine to find the optimal hyperplanes that separate them. These optimal hyperplanes are then aligned by the principal components of their normal vectors. It is proved that the aligned normal vectors provide an unbiased, $\sqrt{n}$-consistent, and asymptotically normal estimator of the sufficient dimension reduction space. The method is then generalized to nonlinear sufficient dimension reduction using the reproducing kernel Hilbert space. In that context, the aligned normal vectors become functions and it is proved that they are unbiased in the sense that they are functions of the true nonlinear sufficient predictors. We compare PSVM with other sufficient dimension reduction methods by simulation and in real data analysis, and through both comparisons firmly establish its practical advantages.