Discrete and Continuous Difference of Submodular Minimization

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.