On the optimality of sliced inverse regression in high dimensions

The central subspace of a pair random variables $(y,\boldsymbol{x})\in \mathbb{R}^{p+1}$ is the minimal $\mathcal{S}$ such that $y\perp\!\!\!\!\!\perp \boldsymbol{x}|P_{\mathcal{S}}\boldsymbol{x}$. In this paper, we consider minimax rate estimating space under 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 $\boldsymbol{\Sigma }$. We first introduce large models depending on smallest nonzero eigenvalue $\lambda $ $\operatorname{var}(\mathbb{E}[\boldsymbol{x}|y])$, over which show an aggregated estimator based SIR procedure converges $d\wedge ((sd+s\log (ep/s))/(n\lambda ))$. then optimal in two scenarios, single and fixed dimension $d$ $. By assuming technical conjecture, can also bounded space.