Independence Properties of Generalized Submodular Information Measures

TL;DR

This paper explores independence properties of generalized submodular information measures, including entropy-based measures, and develops algorithms for optimizing independent set selection with potential applications in combinatorial optimization.

Contribution

It introduces new notions of independence for submodular information measures, analyzes entropy-based independence, and proposes algorithms for finding independent sets.

Findings

Derived independence properties for entropy of random variable sets

Established connections between different notions of independence

Developed algorithms for optimizing independent set selection

Abstract

Recently a class of generalized information measures was defined on sets of items parametrized by submodular functions. In this paper, we propose and study various notions of independence between sets with respect to such information measures, and connections thereof. Since entropy can also be used to parametrize such measures, we derive interesting independence properties for the entropy of sets of random variables. We also study the notion of multi-set independence and its properties. Finally, we present optimization algorithms for obtaining a set that is independent of another given set, and also discuss the implications and applications of combinatorial independence.