A1272
Title: Sliced inverse regression with large structural dimension
Authors: Dongming Huang - National University of Singapore (China)
Songtao Tian - Tsinghua University (China)
Qian Lin - Tisnghua University (China) [presenting]
Abstract: The central space of a joint distribution $(X, Y)$ is the minimal subspace $\mathcal S$ such that $Y\perp\perp X \mid P_{\mathcal S}X$ where $P_{\mathcal S}$ is the projection onto $\mathcal S$. Sliced inverse regression (SIR), one of the most popular methods for estimating the central space, often performs poorly when the structural dimension $d=dim\left( \mathcal S \right)$ is large (e.g., $\geq 5$). It is demonstrated that the generalized signal-noise-ratio (gSNR) tends to be extremely small for a general multiple-index model when $d$ is large. Then the minimax rate for estimating the central space is determineovered a large class of high dimensional distributions with large structural dimension $d$ (i.e., there is no constant upper bound on $d$)in the low gSNR regime. This result extends the existing minimax rate results for estimating the central space of distributions with fixed $d$ to that with large $d$ and clarifies that the decay of signal strength causes the degradation in SIR performance. The technical tools developed here might be of independent interest for studying other central space estimation methods.
