Discrete and Continuous Difference of Submodular Minimization

TL;DR

This paper extends the study of submodular function minimization to both discrete and continuous domains, proposing a novel algorithm with practical applications in integer compressive sensing and least squares.

Contribution

It introduces a unified approach for DS minimization on discrete and continuous domains, including a new variant of the DC Algorithm with theoretical guarantees.

Findings

The proposed method outperforms baselines in experiments.

All functions on discrete domains are DS, and smooth functions on continuous domains are DS.

The algorithm is applicable to continuous domains via discretization.

Abstract

Submodular functions, defined on continuous or discrete domains, arise in numerous applications. We study the minimization of the difference of two submodular (DS) functions, over both domains, extending prior work restricted to set functions. We show that all functions on discrete domains and all smooth functions on continuous domains are DS. For discrete domains, we observe that DS minimization is equivalent to minimizing the difference of two convex (DC) functions, as in the set function case. We propose a novel variant of the DC Algorithm (DCA) and apply it to the resulting DC Program, obtaining comparable theoretical guarantees as in the set function case. The algorithm can be applied to continuous domains via discretization. Experiments demonstrate that our method outperforms baselines in integer compressive sensing and integer least squares.