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