Learning Laplacian Forms for Graph Signal Processing via the Deformed Laplacian

Learning the graph Laplacian from observed data is one of the most investigated and fundamental tasks in Graph Signal Processing (GSP). Different variants of the Laplacian, such as the combinatorial, signless or signed Laplacians have been considered depending on the type of features to be extracted from the data. The main contribution of this paper is the introduction of a parametric Laplacian, called the deformed Laplacian, defined as a quadratic matrix polynomial that provides a parametric dictionary for graph signal processing. The deformed Laplacian can be interpreted as the generator of a parametric linear reaction-diffusion dynamics on graphs, capturing the interplay between diffusive coupling and nodal reaction effects. It is a parametric polynomial matrix that enables the design of novel topological operators tailored to both the underlying graph structure and the observed signals. Interestingly, we show that several Laplacian variants proposed in the literature arise as special cases of the deformed Laplacian. We then develop a method to jointly learn the deformed Laplacian and the graph signals from data, showing how its use improves signal representation across a broad class of graphs compared to standard Laplacian forms. Through extensive numerical experiments on both synthetic and real-world datasets, including financial and communication networks, we assess the benefits of the proposed method in terms of graph signal reconstruction error and sparsity of the representation.

Multimodal Higher-Order Brain Networks: A Topological Signal Processing Perspective

Brain connectomics is still largely dominated by pairwise-based models, such as graphs, which cannot represent circulatory or higher-order functional interactions. In this paper, we propose a multimodal framework based on Topological Signal Processing (TSP) that models the brain as a higher-order topological domain and treats functional interactions as discrete vector fields. We integrate diffusion MRI and resting-state fMRI to learn subject-specific brain cell complexes, where statistically validated structural connectivity defines a sparse scaffold and phase-coupling functional edge signals drive the inference of higher-order interactions (HOIs). Using Hodge-theoretic tools, spectral filtering, and sparse signal representations, our framework disentangles brain connectivity into divergence (source-sink organization), gradient (potential-driven coordination), and curl (circulatory HOIs), enabling the characterization of temporal dynamics through the lens of discrete vector calculus. Across 100 healthy young adults from Human Connectome Project, node-based HOIs are highly individualized, yet robust mesoscale structure emerges under functional-system aggregation. We identify a distributed default mode network-centered gradient backbone and limbic-centered rotational flows; divergence polarization and curl profiles defining circulation regimes with insightful occupancy and dwell-time statistics. These topological signatures yield significant brain-behavior associations, revealing a relevant higher-order organization intrinsic to edge-based models. By making divergence, circulation, and recurrent mesoscale coordination directly measurable, this work enables a principled and interpretable topological phenotyping of brain function.

Topological Signal Processing for 3D Point Cloud Data

Our goal in this paper is to apply the topological signal processing (TSP) framework to the analysis of 3D Point Clouds (PCs) represented on simplicial complexes. Building on Discrete Exterior Calculus (DEC) theory for vector fields, we introduce higher-order Laplacian operators that enable the processing of signals over triangular meshes. Unlike traditional approaches, the proposed approach allows us to characterize both color attributes, modeled as 3D vectors on nodes, and geometry, modeled as 3D vectors on the barycenter of each triangle. Then, we show as TSP tools may efficiently be used to sample, recover and filter PCs attributes treating them as edge signals. Numerical results on synthetic PCs demonstrate accurate color reconstruction with robustness to sparse data and geometry refinement in the case of noisy PC coordinates. The proposed approach provides a topology-based representation to characterize the geometry and attributes of PCs.

Physics-Informed Topological Signal Processing for Water Distribution Network Monitoring

Water management is one of the most critical aspects of our society, together with population increase and climate change. Water scarcity requires a better characterization and monitoring of Water Distribution Networks (WDNs). This paper presents a novel framework for monitoring Water Distribution Networks (WDNs) by integrating physics-informed modeling of the nonlinear interactions between pressure and flow data with Topological Signal Processing (TSP) techniques. We represent pressure and flow data as signals defined over a second-order cell complex, enabling accurate estimation of water pressures and flows throughout the entire network from sparse sensor measurements. By formalizing hydraulic conservation laws through the TSP framework, we provide a comprehensive representation of nodal pressures and edge flows that incorporate higher-order interactions captured through the formalism of cell complexes. This provides a principled way to decompose the water flows in WDNs in three orthogonal signal components (irrotational, solenoidal and harmonic). The spectral representations of these components inherently reflect the conservation laws governing the water pressures and flows. Sparse representation in the spectral domain enable topology-based sampling and reconstruction of nodal pressures and water flows from sparse measurements. Our results demonstrate that employing cell complex-based signal representations enhances the accuracy of edge signal reconstruction, due to proper modeling of both conservative and non-conservative flows along the polygonal cells.

Learning Higher-Order Interactions in Brain Networks via Topological Signal Processing

Our goal in this paper is to leverage the potential of the topological signal processing (TSP) framework for analyzing brain networks. Representing brain data as signals over simplicial complexes allows us to capture higher-order relationships within brain regions of interest (ROIs). Here, we focus on learning the underlying brain topology from observed neural signals using two distinct inference strategies. The first method relies on higher-order statistical metrics to infer multiway relationships among ROIs. The second method jointly learns the brain topology and sparse signal representations, of both the solenoidal and harmonic components of the signals, by minimizing the total variation along triangles and the data-fitting errors. Leveraging the properties of solenoidal and irrotational signals, and their physical interpretations, we extract functional connectivity features from brain topologies and uncover new insights into functional organization patterns. This allows us to associate brain functional connectivity (FC) patterns of conservative signals with well-known functional segregation and integration properties. Our findings align with recent neuroscience research, suggesting that our approach may offer a promising pathway for characterizing the higher-order brain functional connectivities.

Cross-Laplacians Based Topological Signal Processing over Cell MultiComplexes

The study of the interactions among different types of interconnected systems in complex networks has attracted significant interest across many research fields. However, effective signal processing over layered networks requires topological descriptors of the intra- and cross-layers relationships that are able to disentangle the homologies of different domains, at different scales, according to the specific learning task. In this paper, we present Cell MultiComplex (CMC) spaces, which are novel topological domains for representing multiple higher-order relationships among interconnected complexes. We introduce cross-Laplacians matrices, which are algebraic descriptors of CMCs enabling the extraction of topological invariants at different scales, whether global or local, inter-layer or intra-layer. Using the eigenvectors of these cross-Laplacians as signal bases, we develop topological signal processing tools for CMC spaces. In this first study, we focus on the representation and filtering of noisy flows observed over cross-edges between different layers of CMCs to identify cross-layer hubs, i.e., key nodes on one layer controlling the others.

Stability of Graph Convolutional Neural Networks through the lens of small perturbation analysis

In this work, we study the problem of stability of Graph Convolutional Neural Networks (GCNs) under random small perturbations in the underlying graph topology, i.e. under a limited number of insertions or deletions of edges. We derive a novel bound on the expected difference between the outputs of unperturbed and perturbed GCNs. The proposed bound explicitly depends on the magnitude of the perturbation of the eigenpairs of the Laplacian matrix, and the perturbation explicitly depends on which edges are inserted or deleted. Then, we provide a quantitative characterization of the effect of perturbing specific edges on the stability of the network. We leverage tools from small perturbation analysis to express the bounds in closed, albeit approximate, form, in order to enhance interpretability of the results, without the need to compute any perturbed shift operator. Finally, we numerically evaluate the effectiveness of the proposed bound.

Learning Multi-Frequency Partial Correlation Graphs

Despite the large research effort devoted to learning dependencies between time series, the state of the art still faces a major limitation: existing methods learn partial correlations but fail to discriminate across distinct frequency bands. Motivated by many applications in which this differentiation is pivotal, we overcome this limitation by learning a block-sparse, frequency-dependent, partial correlation graph, in which layers correspond to different frequency bands, and partial correlations can occur over just a few layers. To this aim, we formulate and solve two nonconvex learning problems: the first has a closed-form solution and is suitable when there is prior knowledge about the number of partial correlations; the second hinges on an iterative solution based on successive convex approximation, and is effective for the general case where no prior knowledge is available. Numerical results on synthetic data show that the proposed methods outperform the current state of the art. Finally, the analysis of financial time series confirms that partial correlations exist only within a few frequency bands, underscoring how our methods enable the gaining of valuable insights that would be undetected without discriminating along the frequency domain.

Generalized Simplicial Attention Neural Networks

The aim of this work is to introduce Generalized Simplicial Attention Neural Networks (GSANs), i.e., novel neural architectures designed to process data defined on simplicial complexes using masked self-attentional layers. Hinging on topological signal processing principles, we devise a series of self-attention schemes capable of processing data components defined at different simplicial orders, such as nodes, edges, triangles, and beyond. These schemes learn how to weight the neighborhoods of the given topological domain in a task-oriented fashion, leveraging the interplay among simplices of different orders through the Dirac operator and its Dirac decomposition. We also theoretically establish that GSANs are permutation equivariant and simplicial-aware. Finally, we illustrate how our approach compares favorably with other methods when applied to several (inductive and transductive) tasks such as trajectory prediction, missing data imputation, graph classification, and simplex prediction.

Topological Signal Processing over Weighted Simplicial Complexes

Weighing the topological domain over which data can be represented and analysed is a key strategy in many signal processing and machine learning applications, enabling the extraction and exploitation of meaningful data features and their (higher order) relationships. Our goal in this paper is to present topological signal processing tools for weighted simplicial complexes. Specifically, relying on the weighted Hodge Laplacian theory, we propose efficient strategies to jointly learn the weights of the complex and the filters for the solenoidal, irrotational and harmonic components of the signals defined over the complex. We numerically asses the effectiveness of the proposed procedures.