On the optimality of sliced inverse regression in high dimensions

The central subspace of a pair of random variables (y, x) ∈ Rp+¹ is the minimal subspace S such that y x|P_Sx. In this paper, we consider the minimax rate of estimating the central space under the multiple index model y = f (β^τ₁ x, β^τ₂ x,..., β^τ_dx, ε) with at most s active predictors, where x ∼ N(0, Σ) for some class of Σ. We first introduce a large class of models depending on the smallest nonzero eigenvalue λ of var(E[x|y]), over which we show that an aggregated estimator based on the SIR procedure converges at rate d ∧ ((sd + s log(ep/s))/(nλ)). 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 λ. 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.