Discrete and Continuous Difference of Submodular Minimization

2025-06-09

AI Summary

This paper addresses the minimization of the difference of two submodular (DS) functions over both discrete and continuous domains—extending classical DS optimization beyond set functions to general discrete lattices (e.g., integer lattices) and smooth continuous spaces. Methodologically, we propose a novel variant of the DC (Difference-of-Convex) algorithm with theoretical convergence guarantees and broad domain applicability, integrating submodular analysis, DC programming, and discretization-based approximation techniques. Theoretically, we prove that any discrete function and any differentiable convex or concave function can be expressed as a DS decomposition, establishing the universality of the DS framework. Empirically, our method achieves significant improvements over state-of-the-art baselines on structured integer optimization tasks—including integer compressed sensing and integer least squares—demonstrating its efficacy and robustness. Overall, this work provides a unified, principled, and computationally efficient paradigm for structured integer optimization.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• Minimizing difference of submodular functions in discrete and continuous domains
• Extending prior work from set functions to broader domains
• Proposing a novel DC Algorithm variant for improved performance

Innovation

• Extends DS minimization to continuous and discrete domains
• Proposes novel DCA variant for DC Program
• Applies discretization for continuous domain optimization