On the optimality of sliced inverse regression in high dimensions
Qian Lin, Xinran Li, Dongming Huang, Jun S. Liu
The Annals of Statistics · 2021 · Volume 49 · Issue 1
Abstract
The central subspace of a pair of random variables $(y,\boldsymbol{x})\in \mathbb{R}^{p+1}$ is the minimal subspace $\mathcal{S}$ such that $y\perp\!\!\!\!\!\perp \boldsymbol{x}|P_{\mathcal{S}}\boldsymbol{x}$. In this paper, we consider the minimax rate of estimating the central space under the multiple index model $y=f(\boldsymbol{\beta }_{1}^{\tau }\boldsymbol{x},\boldsymbol{\beta }_{2}^{\tau }\boldsymbol{x},\ldots,\boldsymbol{\beta }_{d}^{\tau }\boldsymbol{x},\epsilon )$ with at most $s$ active predictors, where $\boldsymbol{x}\sim N(0,\boldsymbol{\Sigma })$ for some class of $\boldsymbol{\Sigma }$. We first introduce a large class of models depending on the smallest nonzero eigenvalue $\lambda $ of $\operatorname{var}(\mathbb{E}[\boldsymbol{x}|y])$, over which we show that an aggregated estimator based on the SIR procedure converges at rate $d\wedge ((sd+s\log (ep/s))/(n\lambda ))$. 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.
Keywords
62H25, 62J02, Optimal rates, semidefinite positive programming, sliced inverse regression, sufficient dimension reduction
Enable JavaScript for citation tools and additional paper interactions.