Robust Estimation for an Inverse Problem Arising in Multiview Geometry
Journal of Mathematical Imaging and Vision
Non-convex penalized estimation in high-dimensional models with single-index structure
Journal of Multivariate Analysis
Optimal transformation: A new approach for covering the central subspace
Journal of Multivariate Analysis
The Journal of Machine Learning Research
Hi-index | 0.00 |
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,...,βL, 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 maxl=1,...,L βlT (I-A)βl subject to A ∈ Am where Am 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 βl for estimating the EDR subspace. The empirical behavior of the algorithm is studied through numerical simulations.