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

## TL;DR

This paper explores the combinatorial properties of distance covariance as a quasi-concave set function and develops greedy algorithms to find diverse, minimally overlapping feature subsets for regression and classification tasks.

## Contribution

It introduces a novel theoretical framework for distance covariance as a quasi-concave set function and proposes greedy algorithms for diverse feature selection based on this property.

## Key findings

- The negative sample distance covariance function is quasi-concave for dependent variables.
- The proposed greedy algorithms identify all inclusion-minimal maximizers of diversity.
- Application to feature selection yields diverse, relevant variable subsets for predictive modeling.

## 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 [7] to produce subsets of covariates that are both relevant yet ordered in non-increasing levels of diversity of these subsets.