title: Optimal Feature Selection in High-Dimensional Discriminant Analysis.
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 ℓ2convergence 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 (Mai et al., 2012). 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 sizen, dimensionalityp, and sparsity levelsin 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 (Fan et al., 2012) and the sparse optimal scaling estimator (Clemmensen et al., 2011). 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.
authors.affiliation: Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA 15217, USA.
authors.forename: Mladen
authors.initials: M
authors.lastname: Kolar
authors.affiliation: Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USA; Research supported by NSF Grant IIS-1116730.
authors.forename: Han
authors.initials: H
authors.lastname: Liu
keywords: discriminant analysis; high-dimensional statistics; optimal rates of convergence; variable selection