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.

翻译：本研究旨在解决切片逆回归在中心子空间结构维度$d$超过4时降维效果不佳这一长期难题。首先证明在多重指标模型$Y=f( \mathbf{P} \boldsymbol{X})+\epsilon$中（其中$\boldsymbol{X}$为$p$维标准正态向量，$\epsilon$为独立噪声，$\mathbf{P}$为$\mathbb R^{p}$到$\mathbb R^{d}$的投影算子），若链接函数$f$服从高斯过程规律，则在大概率条件下条件期望协方差矩阵$\mathrm{Cov}\left[\mathbb{E}(\boldsymbol{X}\mid Y)\right]$的第$d$个特征值$\lambda_{d}$满足$\lambda_{d}\leq C e^{-\theta d}$（$C$和$\theta$为正常数）。继而聚焦于$\lambda_{d}$可任意小且不超过$d^{-8.1}$的低信号区域，证明中心子空间估计的极小极大风险下界为$\frac{dp}{n\lambda_{d}}$。综合这两项结果，我们为SIR在大维度$d$条件下表现不佳的现象提供了令人信服的解释，这一现象已困扰研究者近三十年。本文发展的技术工具可能对研究其他充分降维方法具有独立价值。