Variable selection of linear programming discriminant estimator

ABSTRACT

In this paper, the variable selection property will be studied for the linear programming discriminant (LPD) estimator, denoted by with n being the sample size. The LPD estimator is used in high-dimensional linear discriminant analysis under the assumption that the Bayes direction is sparse which has support T. More exactly, we will study the property as n → ∞, which means sign consistency. A sufficient condition will be proposed under which the sign consistency property holds as log (p) ⩽ cn for small enough c > 0. The result is also non asymptotic. Our result gives optimal bounds on n and min _{a ∈ T}|β_a| and an optimal bound on .

KEYWORDS:

• Linear classification
• sparsity
• sign consistency
• irrepresentable condition

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62H30
• Secondary 62F12

Acknowledgments

I would like to thank my advisor, Vladimir Koltchinskii, and Karim Lounici for stimulating comments and the patience and time they devoted to me. I would also want to thank an anonymous reviewer for helpful suggestions.

Funding

This work was supported in part by the NSF Grant DMS-1207808.