Structured Connection Graph Learning (SCGL), a block-coordinate algorithm that combines closed-form updates, manifold projections, and spectral constraints, and converges to stationary points of the resulting nonconvex problem, is developed.
Abstract
Connection graphs (CGs) extend classical graphs by associating vector-valued signals to nodes and orthogonal transport maps across edges, making them a natural model for synchronization and manifold-based signal processing. Despite their growing use, learning CGs directly from observations remains challenging because the network topology and the underlying geometric structure are coupled through non-Euclidean orthogonality constraints. In this work, we address this inverse problem by learning a consistent connection graph from noisy vector-valued signals. Exploiting the spectral characterization of consistent CGs, we formulate a structured learning problem that jointly estimates a denoised signal, the graph topology, and node-wise local reference frames. The proposed formulation couples the spectrum of the learned connection Laplacian to that of an underlying combinatorial Laplacian, enabling explicit spectral and topological priors while guaranteeing a nontrivial global-section space. We develop Structured Connection Graph Learning (SCGL), a block-coordinate algorithm that combines closed-form updates, manifold projections, and spectral constraints, and converges to stationary points of the resulting nonconvex problem. Numerical experiments show that SCGL improves topology and geometry recovery over competing approaches, while also yielding effective denoising and signal-compression bases.
We introduce a framework for graph signal processing (GSP) in which signals are represented as graph distribution-valued signals (GDSs), i.e., probability measures in a Wasserstein space. This perspective addresses fundamental limitations of classical vector-based GSP, including the requirement for complete synchronous observations across vertices and the need for strict temporal correspondence in observed filter input--output pairs. Furthermore, by modeling the graph structure as a distribution conditioned on signal realizations, we provide a principled approach to signal-dependent graph structures, which are common in real-world applications, while explicitly encoding uncertainty in graph topology. Our framework inherently captures uncertainty and stochasticity while strictly generalizing traditional graph signals, which can be interpreted as Dirac delta measures. We develop a systematic correspondence between foundational GSP concepts and their GDS analogs, showing that classical formulations emerge as special cases of our framework. We establish theoretical continuity results for GDS transforms, providing stability guarantees for input perturbations and distribution approximations. We demonstrate the utility of this approach through example applications, including graph filter learning and anomaly detection, and validate its effectiveness through empirical studies.
Yanan Zhao, Feng Ji, Xingchao Jian et al.· 0 citations
Equivariant graph neural networks provide a principled way to model geometric systems, but efficient first-order architectures remain limited in how vector information can be transformed as it moves across a graph. We introduce \textsc{ESNN}, an Equivariant Sheaf Neural Network that enriches this interaction by learning directed, matrix-valued transport between neighboring vector features while preserving exact Euclidean equivariance. Rather than increasing the order of the representation, ESNN keeps scalar and vector features first-order and places the additional geometric flexibility in the edge transport itself. We characterize this transport theoretically, showing that when relative displacement is the only covariant geometric input, every linear $O(n)$-equivariant map decomposes into independent radial and tangential components, while learned covariant features enable richer feature-conditioned transformations. We also introduce controlled symmetry relaxation for systems with a preferred ambient direction, which may be prescribed or inferred from data while recovering full $E(n)$-equivariance when the directional pathway is inactive. Across particle dynamics, mesh-based simulation, point-cloud classification, and molecular property prediction, ESNN improves dynamics prediction, recovers the gravity axis when symmetry is broken, yields substantial gains on selected mesh tasks and long-horizon rollouts, and remains robust to unseen rotations. These results show that learning how geometric information is transported across edges offers a complementary route to expressive equivariant message passing without requiring higher-order representations.
Alessio Borgi, M. Severino, Fabrizio Silvestri et al.· 0 citations
Laplacian-regularized minimization is fundamental in signal processing and machine learning, but is limited by the dense and ill-conditioned nature of the graph Laplacian pseudoinverse. While the Laplacian itself is sparse, its pseudoinverse is dense and often ill-conditioned, rendering direct computation impractical at scale. Moreover, pseudoinverse learning is more challenging than Laplacian learning. To address this challenge, this paper considers the setting where the graph Laplacian is given and proposes a Difference-of-Convex Regularizer (DCR) graph learning framework that approximates the spectral action of the Laplacian pseudoinverse without direct inversion via regularized Maximum Likelihood Estimation (MLE). By reformulating Laplacian-Regularized Nonnegative Least Squares (LR-NNLS) through a dual representation, DCR decouples pseudoinverse learning from instance-specific inference and enables efficient primal solution reconstruction via a differentiable dual-guided learning scheme. We establish theoretical guarantees on stability and the existence of a unique fixed point for DCR algorithm. Numerical experiments demonstrate improved performance over convex solvers and graph filtering baselines and robust performance across diverse graph topologies.
Modern sensing, communication, and learning systems generate heterogeneous network signals, with local data differing in dimension, modality, and geometric structure. Processing such data requires a mathematical framework capable of simultaneously modeling heterogeneous local signal spaces and the transformations relating them. Network sheaves provide such a framework by associating local vector spaces with network entities and linear restriction maps with their interactions. This is the first paper to develop a unified sheaf signal processing (SSP) framework on network sheaves, extending the fundamental operations of signal processing, namely spectral analysis, filtering, and sampling, to heterogeneous local spaces. Unlike graph and topological signal processing, where signals are modeled over a common vector space, SSP jointly models heterogeneous local signal spaces and the linear transformations relating neighboring spaces through restriction maps. We define the Sheaf Fourier Transform (SFT), whose frequencies quantify signal inconsistency induced by the network topology, the restriction maps, and the local geometry. Building on this representation, we develop polynomial sheaf filters and formulate sampling as the joint selection of network nodes and intra-node components. We derive perfect recovery conditions for bandlimited sheaf signals and propose a greedy sampling-set design algorithm. To incorporate application-dependent signal models, including different bases, dictionaries, and learned embeddings, we introduce representation sheaves and characterize the natural transformations that preserve spectral properties and guarantee interoperability across representations. Experiments on synthetic, motion-capture, and financial datasets validate the proposed framework and demonstrate consistent improvements over canonical graph signal processing baselines.
Gabriele D’Acunto, Leonardo Di Nino, P. Lorenzo et al.· 0 citations
Graph machine learning tailored to planar graphs is developed, with an emphasis on the mathematical intuition that connects the topology of a plane embedding to the spectral and combinatorial structure ex- ploited by learning algorithms.
Satyanarayana Sanakkayala· International Journal of Com...· 0 citations
This work develops this perspective theoretically, showing that quotient couplings can be lifted to aligned representatives without additional cost and that symmetrization yields equivariant flow-matching minimizers, including for categorical endpoint prediction.