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

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 $P_{\mathbf{X}}$ being elliptical symmetric). In this paper, we show that a fast parametric convergence rate of form $C_d \cdot n^{-1/2}$ is achievable via estimating the \emph{expected smoothed gradient outer product}, for a general class of distribution $P_{\mathbf{X}}$ admitting Gaussian or heavier distributions. When the link function is a polynomial with a degree of at most $r$ and $P_{\mathbf{X}}$ is the standard Gaussian, we show that the prefactor depends on the ambient dimension $d$ as $C_d \propto d^r$.