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.

Neural Information Processing Systems http://nips.cc/

We thank the reviewers for their valuable comments and for acknowledging the novelty of our work. We hope to address adequately the concerns raised by the reviewers. We apologize for overlooking this. The major computational complexity of our algorithm is the eigenvalue decomposition. ' could be determined by some

Learning graphs from data is an important problem finding many practical applications. This paper contributes by a new regularization strategy that employs spectral constraints of graph Laplacian. The resulting algorithm appears to be novel and technically sound. However, I think this paper will benefit significantly from major revision making the practical relevance of the proposed approach clearer: 1) The authors originally suggested four different spectral constraints but only one of them (k-component graph) was actually evaluated. Implementing and evaluating more than one constraints in this framework could help understand the nature of this strategy.

Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. It is well known that structured graph learning from observed samples is an NP-hard combinatorial problem. In this paper, we first show that for a set of important graph families it is possible to convert the structural constraints of structure into eigenvalue constraints of the graph Laplacian matrix. Then we introduce a unified graph learning framework, lying at the integration of the spectral properties of the Laplacian matrix with Gaussian graphical modeling that is capable of learning structures of a large class of graph families. The proposed algorithms are provably convergent and practically amenable for large-scale semi-supervised and unsupervised graph-based learning tasks. Extensive numerical experiments with both synthetic and real data sets demonstrate the effectiveness of the proposed methods. An R package containing code for all the experimental results is available at https://cran.r-project.org/package=spectralGraphTopology.

Graph learning from data represents a canonical problem that has received substantial attention in the literature. However, insufficient work has been done in incorporating prior structural knowledge onto the learning of underlying graphical models from data. Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. Useful structured graphs include the multi-component graph, bipartite graph, connected graph, sparse graph, and regular graph. In general, structured graph learning is an NP-hard combinatorial problem, therefore, designing a general tractable optimization method is extremely challenging. In this paper, we introduce a unified graph learning framework lying at the integration of Gaussian graphical models and spectral graph theory. To impose a particular structure on a graph, we first show how to formulate the combinatorial constraints as an analytical property of the graph matrix. Then we develop an optimization framework that leverages graph learning with specific structures via spectral constraints on graph matrices. The proposed algorithms are provably convergent, computationally efficient, and practically amenable for numerous graph-based tasks. Extensive numerical experiments with both synthetic and real data sets illustrate the effectiveness of the proposed algorithms. The code for all the simulations is made available as an open source repository.