On the optimality of sliced inverse regression in high dimensions

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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.