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Efficient Estimation of the Central Mean Subspace via Smoothed Gradient Outer Products
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DOI：10.1137/23M1626700.png)
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Abstract
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We consider the problem of sufficient dimension reduction for multi-index models. The estimators of the central mean subspace in prior works either have slow (nonparametric) convergence rates or rely on stringent distributional conditions (e.g., elliptical symmetric covariate distribution P\bfitX). In this paper, we show that a fast parametric convergence rate of form Cd \cdot n-1/2 is achievable via estimating the expected smoothed gradient outer product for a general class of distribution P\bfitX that admits Gaussian or heavier distributions. When the link function is a polynomial with a degree of at most r and P\bfitX is the standard Gaussian, we show that the prefactor depends on the ambient dimension d as Cd \propto dr.
Keywords:
central mean subspace
multi-index model
smoothed gradient outer product
sufficient dimension reduction
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