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In this paper we show that the negative sample distance covariance functionis a quasi-concave set function of samples of random variables that are notstatistically independent. We use these properties to propose greedy algorithmsto combinatorially optimize some diversity (low statistical dependence)promoting functions of distance covariance. Our greedy algorithm obtains allthe inclusion-minimal maximizers of this diversity promoting objective.Inclusion-minimal maximizers are multiple solution sets of globally optimalmaximizers that are not a proper subset of any other maximizing set in thesolution set. We present results upon applying this approach to obtain diversefeatures (covariates/variables/predictors) in a feature selection setting forregression (or classification) problems. We also combine our diverse featureselection algorithm with a distance covariance based relevant feature selectionalgorithm of [7] to produce subsets of covariates that are both relevant yetordered in non-increasing levels of diversity of these subsets.

Combinatorics of Distance Covariance: Inclusion-Minimal Maximizers of Quasi-Concave Set Functions for Diverse Variable Selection