TY - JOUR

T1 - On the optimality of sliced inverse regression in high dimensions

AU - Lin, Qian

AU - Li, Xinran

AU - Huang, Dongming

AU - Liu, Jun S.

N1 - Funding Information:
Acknowledgments. The authors thank the Associate Editor and three referees for their constructive comments. Lin’s research is supported in part by the NSFC (Grant 11971257), Beijing NSF (Grant Z190001) and Beijing Academy of Artificial Intelligence. Liu’s research is supported in part by NIH Grant R01 GM113242-01, NSF grants DMS-1613035 and DMS-1903139.
Publisher Copyright:
© Institute of Mathematical Statistics, 2021

PY - 2021/2

Y1 - 2021/2

N2 - The central subspace of a pair of random variables (y, x) ∈ Rp+1 is the minimal subspace S such that y x|PSx. In this paper, we consider the minimax rate of estimating the central space under the multiple index model y = f (βτ1 x, βτ2 x,..., βτdx, 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.

AB - The central subspace of a pair of random variables (y, x) ∈ Rp+1 is the minimal subspace S such that y x|PSx. In this paper, we consider the minimax rate of estimating the central space under the multiple index model y = f (βτ1 x, βτ2 x,..., βτdx, 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.

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U2 - 10.1214/19-AOS1813

DO - 10.1214/19-AOS1813

M3 - Article

AN - SCOPUS:85101243286

VL - 49

SP - 1

EP - 20

JO - Annals of Statistics

JF - Annals of Statistics

SN - 0090-5364

IS - 1

ER -