  The central subspace of a pair of random variables $(y,x) \in
\mathbb{R}^{p+1}$ is the minimal subspace $\mathcal{S}$ such that $y \perp
\hspace{-2mm} \perp x\mid P_{\mathcal{S}}x$. In this paper, we consider the
minimax rate of estimating the central space of the multiple index models
$y=f(\beta_{1}^{\tau}x,\beta_{2}^{\tau}x,...,\beta_{d}^{\tau}x,\epsilon)$ with
at most $s$ active predictors where $x \sim N(0,I_{p})$. We first introduce a
large class of models depending on the smallest non-zero eigenvalue $\lambda$
of $var(\mathbb{E}[x|y])$, over which we show that an aggregated estimator
based on the SIR procedure converges at rate
$d\wedge((sd+s\log(ep/s))/(n\lambda))$. We then show that this rate is optimal
in two scenarios: the single index models; and the multiple index models with
fixed central dimension $d$ and fixed $\lambda$. By assuming a technical
conjecture, we can show that this rate is also optimal for multiple index
models with bounded dimension of the central space. We believe that these
(conditional) optimal rate results bring us meaningful insights of general SDR
problems in high dimensions.
