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Learning the Structure of Connection Graphs

Leonardo Di Nino, Gabriele D'Acunto, Sergio Barbarossa, Paolo Di Lorenzo

TL;DR

The paper tackles learning connection graphs by enforcing spectral consistency between the connection Laplacian $\mathbb{L}$ and the underlying combinatorial Laplacian $\mathbf{L}$ via a consistency condition. It introduces Structured Connection Graph Learning (SCGL), a nonconvex, geometry-aware framework solved with alternating block descent on manifolds, jointly estimating edge weights, node-wise orthogonal maps, and the Kronecker-structured Laplacian $\mathcal{L}_{\mathbb{K}}(\mathbf{w})$ that governs diffusion on the CG. The method achieves improved topology recovery and geometric fidelity over baselines on synthetic CGs, validating both topology and geometry reconstruction under limited samples. This work enables geometry-aware network inference for applications in synchronization, Riemannian signal processing, and neural sheaf diffusion, with scalable optimization and potential extensions to noisy or real-world data.

Abstract

Connection graphs (CGs) extend traditional graph models by coupling network topology with orthogonal transformations, enabling the representation of global geometric consistency. They play a key role in applications such as synchronization, Riemannian signal processing, and neural sheaf diffusion. In this work, we address the inverse problem of learning CGs directly from observed signals. We propose a principled framework based on maximum pseudo-likelihood under a consistency assumption, which enforces spectral properties linking the connection Laplacian to the underlying combinatorial Laplacian. Based on this formulation, we introduce the Structured Connection Graph Learning (SCGL) algorithm, a block-optimization procedure over Riemannian manifolds that jointly infers network topology, edge weights, and geometric structure. Our experiments show that SCGL consistently outperforms existing baselines in both topological recovery and geometric fidelity, while remaining computationally efficient.

Learning the Structure of Connection Graphs

TL;DR

The paper tackles learning connection graphs by enforcing spectral consistency between the connection Laplacian and the underlying combinatorial Laplacian via a consistency condition. It introduces Structured Connection Graph Learning (SCGL), a nonconvex, geometry-aware framework solved with alternating block descent on manifolds, jointly estimating edge weights, node-wise orthogonal maps, and the Kronecker-structured Laplacian that governs diffusion on the CG. The method achieves improved topology recovery and geometric fidelity over baselines on synthetic CGs, validating both topology and geometry reconstruction under limited samples. This work enables geometry-aware network inference for applications in synchronization, Riemannian signal processing, and neural sheaf diffusion, with scalable optimization and potential extensions to noisy or real-world data.

Abstract

Connection graphs (CGs) extend traditional graph models by coupling network topology with orthogonal transformations, enabling the representation of global geometric consistency. They play a key role in applications such as synchronization, Riemannian signal processing, and neural sheaf diffusion. In this work, we address the inverse problem of learning CGs directly from observed signals. We propose a principled framework based on maximum pseudo-likelihood under a consistency assumption, which enforces spectral properties linking the connection Laplacian to the underlying combinatorial Laplacian. Based on this formulation, we introduce the Structured Connection Graph Learning (SCGL) algorithm, a block-optimization procedure over Riemannian manifolds that jointly infers network topology, edge weights, and geometric structure. Our experiments show that SCGL consistently outperforms existing baselines in both topological recovery and geometric fidelity, while remaining computationally efficient.

Paper Structure

This paper contains 6 sections, 1 theorem, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary definitions
  3. Problem formulation
  4. Structured Connection Graph Learning
  5. Numerical results
  6. Conclusions

Key Result

Theorem 1

Let $\mathbb{G}=\langle \mathcal{G}, \mathbb{R}^n,\mathbf{w}, \mathrm{O}({n})\xspace \rangle$ be a CG having $v$ nodes and connection Laplacian $\mathbb{L} \in \mathbb{R}\xspace^{vn \times vn}$. Let $\mathbf{L}$ be the combinatorial Laplacian of the underlying graph $\mathcal{G}$. Hence, the followi

Figures (2)

  • Figure 1: F1 score (left), empirical total variation (central) and MSE on edge weights (right) for the identification of ER CGs.
  • Figure 2: Graphs retrieved with SCGL and the considered baselines from data generated around a discrete sphere whose ground true connection Laplacian is derived from VDM and spectral synchronization.

Theorems & Definitions (4)

  • Definition 1: Connection graph
  • Definition 2: Connection Laplacian
  • Theorem 1: chung2014ranking
  • Definition 3: Kronecker-structured Laplacian operator