ON THE OPTIMALITY OF SLICED INVERSE REGRESSION IN HIGH DIMENSIONS

• 成果类型: Article
• 署名作者: Lin, Qian; Li, Xinran; Huang, Dongming; Liu, Jun S.
• 署名单位: Tsinghua University; University of Illinois System; University of Illinois Urbana-Champaign; National University of Singapore; Harvard University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1813
• 发表日期: 2021
• 页码: 1-20
• 关键词: optimal rates; sparse pca; Consistency; reduction; selection

摘要

The central subspace of a pair of random variables (y, x) is an element of Rp+1 is the minimal subspace S such that y perpendicular to x vertical bar P(S)x. In this paper, we consider the minimax rate of estimating the central space under the multiple index model y = f(beta(tau)(1) x, beta(tau)(d), ..., beta(tau)(d)x,is an element of) with at most s active predictors, where x similar to N(0, Sigma) for some class of Sigma. We first introduce a large class of models depending on the smallest nonzero eigenvalue lambda of var(E[x vertical bar y]), over which we show that an aggregated estimator based on the SIR procedure converges at rate d Lambda ((sd + s log(ep/s))/(n lambda)). We then show that this rate is optimal in two scenarios, the single index models and the multiple index models with fixed central dimension d and fixed lambda. By assuming a technical conjecture, we can show that this rate is also optimal for multiple index models with bounded dimension of the central space.