Efficient Estimation of the Central Mean Subspace via Smoothed Gradient Outer Products

Abstract

We consider the problem of sufficient dimension reduction (SDR) for multi-index models. The estimators of the central mean subspace in prior works either have slow (non-parametric) convergence rates, or rely on stringent distributional conditions (e.g., the covariate distribution P_X being elliptical symmetric). In this paper, we show that a fast parametric convergence rate of form C_d n^-1/2 is achievable via estimating the expected smoothed gradient outer product, for a general class of distribution P_X admitting Gaussian or heavier distributions. When the link function is a polynomial with a degree of at most r and P_X is the standard Gaussian, we show that the prefactor depends on the ambient dimension d as C_d d^r.

Tasks

Reproductions