## 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
β_{1},...,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

_{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.

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