On the structural dimension of sliced inverse regression
研究了切片逆回归在结构维度超过4时表现不佳的原因，发现其第d个特征值随d指数衰减，且估计中心空间的极小风险有下界，解释了困扰学界近三十年的现象。
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(PX)+ϵ where X is a p-standard normal vector, ϵ is an independent noise, and P is a projection operator from Rp to Rd, if the link function f follows the law of a Gaussian process. Then with high probability, the dth eigenvalue λd of Cov[E(X|Y)] satisfies λd≤Ce−θd for some positive constants C and θ. We then focus on the low signal regime where λd 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 dp nλd. 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.