  In this work, we address the longstanding puzzle that Sliced Inverse
Regression (SIR) often performs poorly for sufficient dimension reduction when
the structural dimension $d$ (the dimension of the central space) exceeds 4. We
first show that in the multiple index model $Y=f( \mathbf{P}
\boldsymbol{X})+\epsilon$ where $\boldsymbol{X}$ is a $p$-standard normal
vector, $\epsilon$ is an independent noise, and $\mathbf{P}$ is a projection
operator from $\mathbb R^{p}$ to $\mathbb R^{d}$, if the link function $f$
follows the law of a Gaussian process, then with high probability, the $d$-th
eigenvalue $\lambda_{d}$ of $\mathrm{Cov}\left[\mathbb{E}(\boldsymbol{X}\mid
Y)\right]$ satisfies $\lambda_{d}\leq C e^{-\theta d}$ for some positive
constants $C$ and $\theta$. We then focus on the low signal regime where
$\lambda_{d}$ can be arbitrarily small and not larger than $d^{-8.1}$, and
prove that the minimax risk of estimating the central space is lower bounded by
$\frac{dp}{n\lambda_{d}}$. Combining these two results, we provide a convincing
explanation for the poor performance of SIR when $d$ is large, a phenomenon
that has perplexed researchers for nearly three decades. The technical tools
developed here may be of independent interest for studying other sufficient
dimension reduction methods.
