On the optimality of sliced inverse regression in high dimensions
On the optimality of sliced inverse regression in high dimensions
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 $s$ …