On Stein's Identity and Near-Optimal Estimation in High-dimensional Index Models
This work addresses a challenge in signal processing, machine learning, and statistics by providing robust estimators that relax common Gaussian assumptions, though it appears incremental as it builds on existing Stein's identity methods.
The paper tackles the problem of estimating parametric components in high-dimensional semi-parametric multiple index models under non-Gaussian and heavy-tailed conditions, achieving near-optimal statistical convergence rates as demonstrated in simulations.
We consider estimating the parametric components of semi-parametric multiple index models in a high-dimensional and non-Gaussian setting. Such models form a rich class of non-linear models with applications to signal processing, machine learning and statistics. Our estimators leverage the score function based first and second-order Stein's identities and do not require the covariates to satisfy Gaussian or elliptical symmetry assumptions common in the literature. Moreover, to handle score functions and responses that are heavy-tailed, our estimators are constructed via carefully thresholding their empirical counterparts. We show that our estimator achieves near-optimal statistical rate of convergence in several settings. We supplement our theoretical results via simulation experiments that confirm the theory.