Efficient Estimation of the Central Mean Subspace via Smoothed Gradient
  Outer Products
Gan Yuan Mingyue Xu Samory Kpotufe Daniel Hsu
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 $C_d \propto d^r$.