Joint Simplicial Complex Learning via Binary Linear Programming

TL;DR

This paper introduces a joint learning framework for higher-order network structures using binary linear programming, effectively capturing multi-way interactions and outperforming existing methods.

Contribution

It proposes a novel linear programming approach that enforces the inclusion property for simplicial complexes, enabling simultaneous estimation of edges and higher-order simplices.

Findings

Outperforms hierarchical and greedy baselines

Faithfully enforces higher-order structural priors

Works on simulated and real-world data

Abstract

Learning the topology of higher-order networks from data is a fundamental challenge in many signal processing and machine learning applications. Simplicial complexes provide a principled framework for modeling multi-way interactions, yet learning their structure is challenging due to the strong coupling across different simplicial levels imposed by the inclusion property. In this work, we propose a joint framework for simplicial complex learning that enforces the inclusion property through a linear constraint, enabling the formulation of the problem as a binary linear program. The objective function consists of a combination of smoothness measures across all considered simplicial levels, allowing for the incorporation of arbitrary smoothness criteria. This formulation enables the simultaneous estimation of edges and higher-order simplices within a single optimization problem. Experiments on simulated and real-world data demonstrate that the proposed joint approach outperforms hierarchical and greedy baselines, while more faithfully enforcing higher-order structural priors.