Simultaneous Sparse Estimation of Canonical Vectors in the p ≫ N Setting

ABSTRACT

This article considers the problem of sparse estimation of canonical vectors in linear discriminant analysis when p ≫ N. Several methods have been proposed in the literature that estimate one canonical vector in the two-group case. However, G − 1 canonical vectors can be considered if the number of groups is G. In the multi-group context, it is common to estimate canonical vectors in a sequential fashion. Moreover, separate prior estimation of the covariance structure is often required. We propose a novel methodology for direct estimation of canonical vectors. In contrast to existing techniques, the proposed method estimates all canonical vectors at once, performs variable selection across all the vectors and comes with theoretical guarantees on the variable selection and classification consistency. First, we highlight the fact that in the N > p setting the canonical vectors can be expressed in a closed form up to an orthogonal transformation. Secondly, we propose an extension of this form to the p ≫ N setting and achieve feature selection by using a group penalty. The resulting optimization problem is convex and can be solved using a block-coordinate descent algorithm. The practical performance of the method is evaluated through simulation studies as well as real data applications. Supplementary materials for this article are available online.

KEYWORDS:

• Block-coordinate descent
• Classification
• Dimension reduction
• Discriminant analysis
• Feature selection
• Group penalization

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Acknowledgments

We are grateful to Jacob Bien for a valuable discussion of block-coordinate descent algorithms. We thank the Editor, the Associate Editor and two anonymous referees for valuable comments and suggestions that helped to significantly improve this manuscript.

Funding

This research was partially supported by NSF-DMS 1208488, NSF-DMS 0808864, NIH grant U19 AI111143 and ASAF grant FA9550-13-1-0137.