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Joint Simplicial Complex Learning via Binary Linear Programming

Varun Sarathchandran, Geert Leus

TL;DR

This work tackles learning the topology of higher-order networks by jointly recovering edges and triangles in simplicial complexes from data. It formulates the problem as a binary linear program that enforces the simplicial inclusion across levels via a linear constraint on full-structure selection variables $\mathbf{s}_1$ and $\mathbf{s}_2$, while incorporating flexible smoothness priors. A novel similarity-based edge-smoothness term is introduced, alongside the traditional curl-based measure, enabling more faithful higher-order topology learning. Empirical results on simulated and real data show that the proposed joint approach outperforms hierarchical and greedy baselines, particularly when the assumed priors match the data, highlighting the method’s practical impact for robust higher-order network inference.

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.

Joint Simplicial Complex Learning via Binary Linear Programming

TL;DR

This work tackles learning the topology of higher-order networks by jointly recovering edges and triangles in simplicial complexes from data. It formulates the problem as a binary linear program that enforces the simplicial inclusion across levels via a linear constraint on full-structure selection variables and , while incorporating flexible smoothness priors. A novel similarity-based edge-smoothness term is introduced, alongside the traditional curl-based measure, enabling more faithful higher-order topology learning. Empirical results on simulated and real data show that the proposed joint approach outperforms hierarchical and greedy baselines, particularly when the assumed priors match the data, highlighting the method’s practical impact for robust higher-order network inference.

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.
Paper Structure (10 sections, 10 equations, 1 figure)

This paper contains 10 sections, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial Complexes and Signals
  4. Simplicial Laplacians and Smoothness
  5. Problem Formulation
  6. Binary Linear Program Formulation
  7. Experiments
  8. Baselines
  9. Numerical Results
  10. Conclusion

Figures (1)

  • Figure 1: The f1 scores of methods on edge and triangle detection on simulated and real data. All methods rely on node signal smoothness to identify edges, and thus labels specify the type of smoothness used to identify triangles. (a) Performance on simulated data with edge signals having low curl on triangles. (b) Performance on simulated data with similar edge signals on a triangle. (c) Performance of all methods with both smoothness types on real co-authorship data. Results are aggregated over 10 random realizations.