Optimal variable selection in multi-group sparse discriminant analysis

TL;DR

This paper establishes optimal variable selection methods for multi-group sparse discriminant analysis, achieving faster convergence rates and matching minimax optimality known in the two-group case.

Contribution

It provides sharp conditions and optimal convergence rates for variable recovery in multi-group discriminant analysis, improving upon existing methods.

Findings

Achieves optimal sample size scaling for variable selection

Rates of convergence match minimax bounds in the two-group case

Numerical validation confirms theoretical results

Abstract

This article considers the problem of multi-group classification in the setting where the number of variables $p$ is larger than the number of observations $n$. Several methods have been proposed in the literature that address this problem, however their variable selection performance is either unknown or suboptimal to the results known in the two-group case. In this work we provide sharp conditions for the consistent recovery of relevant variables in the multi-group case using the discriminant analysis proposal of Gaynanova et al., 2014. We achieve the rates of convergence that attain the optimal scaling of the sample size $n$, number of variables $p$ and the sparsity level $s$. These rates are significantly faster than the best known results in the multi-group case. Moreover, they coincide with the optimal minimax rates for the two-group case. We validate our theoretical results with numerical analysis.