Joint Simplicial Complex Learning via Binary Linear Programming

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.

💡 Research Summary

This paper tackles the problem of learning the topology of higher‑order networks—specifically simplicial complexes (SCs)—directly from observed signals. While graph (0‑ and 1‑simplex) learning is well‑studied, extending inference to higher‑order simplices such as triangles (2‑simplices) remains difficult because the inclusion property (every face of a simplex must also belong to the complex) tightly couples different simplicial levels. Existing approaches either adopt a hierarchical pipeline (first infer edges, then infer triangles based on the recovered edges) or use a greedy alternating scheme that relaxes the inclusion constraint into a penalty term. Both strategies either break the coupling between levels or fail to guarantee exact inclusion.

The authors propose a joint one‑shot framework that enforces the inclusion property with a linear constraint. Let (s_1 \in {0,1}^{\bar N_1}) be a binary selection vector for candidate edges and (s_2 \in {0,1}^{\bar N_2}) for candidate triangles. The unoriented edge‑to‑triangle incidence matrix (\bar B^{+}_2) counts how many selected triangles contain each edge. By imposing
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