Learning the Structure of Connection Graphs

2025-10-13

AI Summary

This paper addresses the inverse problem of directly inferring the structure of connection graphs (CGs) from observed signals. Existing methods struggle to jointly recover topology, edge weights, and geometric structure. To overcome this, we propose Structured Connection Graph Learning (SCGL): (i) we establish, for the first time, a theoretical link between the spectral properties of the connection Laplacian and those of the combinatorial Laplacian; (ii) under a consistency assumption, we formulate a joint optimization framework based on maximum pseudo-likelihood estimation with spectral constraints; and (iii) we design an efficient block-coordinate optimization algorithm operating on the Riemannian manifold of orthogonal matrices. Experiments demonstrate that SCGL significantly outperforms state-of-the-art baselines in both topological accuracy and geometric fidelity, while exhibiting strong generalization capability and computational efficiency.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Web Mining and Content Analysis: Bridging structured and unstructured 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.

Problem

• Learning connection graphs from observed signals directly
• Inferring network topology and geometric structure jointly
• Solving inverse problem using maximum pseudo-likelihood framework

Innovation

• Learning connection graphs from observed signals
• Maximum pseudo-likelihood framework with consistency assumption
• Block-optimization over Riemannian manifolds for joint inference