Optimal Feature Selection in High-Dimensional Discriminant Analysis
This work addresses the need for interpretable models in scientific data analysis by providing rigorous theoretical guarantees for variable selection in high-dimensional settings, though it is incremental as it builds on existing methods like sparse discriminant analysis.
The paper tackles the lack of sharp theoretical analysis for variable selection performance in high-dimensional discriminant analysis by establishing sharp sufficient conditions for consistent variable selection using sparse discriminant analysis, achieving significantly faster convergence rates and optimal scaling of sample size, dimensionality, and sparsity.
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 (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 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 (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.