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.
Categories: math.ST stat.TH