  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 $C_d \propto d^r$.
