Gradient-based kernel dimension reduction for regression

This paper proposes a novel approach to linear dimension reduction for regression using nonparametric estimation with positive definite kernels or reproducing kernel Hilbert spaces. The purpose of the dimension reduction is to find such directions in the explanatory variables that explain the response sufficiently: this is called sufficient dimension reduction. The proposed method is based on an estimator for the gradient of regression function considered for the feature vectors mapped into reproducing kernel Hilbert spaces. It is proved that the method is able to estimate the directions that achieve sufficient dimension reduction. In comparison with other existing methods, the proposed one has wide applicability without strong assumptions on the ∗The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562 Japan †Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK, and Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, Singapore, 117546