Diverse data selection via combinatorial quasi-concavity of distance covariance: A polynomial time global minimax algorithm, Praneeth Vepakomma, Yulia Kempner

Abstract

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 Kong et al. (2015) to produce subsets of covariates that are both relevant yet ordered in non-increasing levels of diversity of these subsets.