A New Algorithm for Estimating the Effective Dimension-Reduction Subspace

Arnak S. Dalalyan; Anatoly Juditsky; Vladimir Spokoiny

A New Algorithm for Estimating the Effective Dimension-Reduction Subspace

Arnak S. Dalalyan, Anatoly Juditsky, Vladimir Spokoiny; 9(53):1647−1678, 2008.

Abstract

The statistical problem of estimating the effective dimension-reduction (EDR) subspace in the multi-index regression model with deterministic design and additive noise is considered. A new procedure for recovering the directions of the EDR subspace is proposed. Many methods for estimating the EDR subspace perform principal component analysis on a family of vectors, say β₁,...,β_L, nearly lying in the EDR subspace. This is in particular the case for the structure-adaptive approach proposed by Hristache et al. (2001a). In the present work, we propose to estimate the projector onto the EDR subspace by the solution to the optimization problem

minimize max_l=1,...,L β_l^T (I-A)β_l subject to A ∈ A_m

where A_m is the set of all symmetric matrices with eigenvalues in [0,1] and trace less than or equal to m, with m being the true structural dimension. Under mild assumptions, √n-consistency of the proposed procedure is proved (up to a logarithmic factor) in the case when the structural dimension is not larger than 4. Moreover, the stochastic error of the estimator of the projector onto the EDR subspace is shown to depend on L logarithmically. This enables us to use a large number of vectors β_l for estimating the EDR subspace. The empirical behavior of the algorithm is studied through numerical simulations.