Optimal feature selection in high-dimensional discriminant analysis

Mladen Kolar, Han Liu

Research output: Contribution to journalArticle

10 Scopus citations

Abstract

We consider the high-dimensional discriminant analysis problem. For this problem, different methods have been proposed and justified by establishing exact convergence rates for the classification risk, as well as the ℓ2 convergence results to the discriminative rule. However, sharp theoretical analysis for the variable selection performance of these procedures have not been established, even though model interpretation is of fundamental importance in scientific data analysis. This paper bridges the gap by providing sharp sufficient conditions for consistent variable selection using the sparse discriminant analysis. Through careful analysis, we establish rates of convergence that are significantly faster than the best known results and admit an optimal scaling of the sample size n, dimensionality p, and sparsity level s in the high-dimensional setting. Sufficient conditions are complemented by the necessary information theoretic limits on the variable selection problem in the context of high-dimensional discriminant analysis. Exploiting a numerical equivalence result, our method also establish the optimal results for the ROAD estimator and the sparse optimal scoring estimator. Furthermore, we analyze an exhaustive search procedure, whose performance serves as a benchmark, and show that it is variable selection consistent under weaker conditions. Extensive simulations demonstrating the sharpness of the bounds are also provided.

Original languageEnglish (US)
Article number6985722
Pages (from-to)1063-1083
Number of pages21
JournalIEEE Transactions on Information Theory
Volume61
Issue number2
DOIs
StatePublished - Feb 1 2015

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

Keywords

  • High-dimensional statistics
  • discriminant analysis
  • optimal rates of convergence
  • variable selection

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