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E. Isufi
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1
Matched subspace detection (MSD) is a powerful tool recently generalized from Euclidean data to graph signal processing. However, existing graph-based MSD methods are often limited by assumptions of known noise variance and by overlooking the statistical properties of the graph Fourier transform (GFT) coefficients thereby limiting practical applicability. To address these gaps, this paper introduces two novel generalized likelihood ratio (GLR) tests for graph-based MSD. The first-order GLR test operates without knowledge of the noise variance and the GFT coefficients by estimating them via maximum likelihood. The second-order GLR test further incorporates a Gaussian prior on the GFT coefficients, yielding a more powerful and comprehensive statistical model. Experimental results demonstrate that our proposed detectors are robust and effective, particularly in challenging noisy scenarios, highlighting their importance for detection tasks in graph signal processing.
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Matched subspace detection (MSD) is a powerful tool recently generalized from Euclidean data to graph signal processing. However, existing graph-based MSD methods are often limited by assumptions of known noise variance and by overlooking the statistical properties of the graph Fourier transform (GFT) coefficients thereby limiting practical applicability. To address these gaps, this paper introduces two novel generalized likelihood ratio (GLR) tests for graph-based MSD. The first-order GLR test operates without knowledge of the noise variance and the GFT coefficients by estimating them via maximum likelihood. The second-order GLR test further incorporates a Gaussian prior on the GFT coefficients, yielding a more powerful and comprehensive statistical model. Experimental results demonstrate that our proposed detectors are robust and effective, particularly in challenging noisy scenarios, highlighting their importance for detection tasks in graph signal processing.
Graph learning aims to infer a network structure directly from observed data, enabling the analysis of complex dependencies in irregular domains. Traditional methods focus on scalar signals at each node, ignoring dependencies along additional dimensions such as time, configurations of the observation device, or populations. In this work, we propose a graph signal processing framework for learning graphs from two-dimensional signals, modeled as matrix graph signals generated by joint filtering along both dimensions. This formulation leverages the concept of graph stationarity across the two dimensions and leverages product graph representations to capture structured dependencies. Based on this model, we design an optimization problem that can be solved efficiently and provably recovers the optimal underlying Kronecker/Cartesian/strong product graphs. Experiments on synthetic data demonstrate that our approach achieves higher estimation accuracy and reduced computational cost compared to existing methods.
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Graph learning aims to infer a network structure directly from observed data, enabling the analysis of complex dependencies in irregular domains. Traditional methods focus on scalar signals at each node, ignoring dependencies along additional dimensions such as time, configurations of the observation device, or populations. In this work, we propose a graph signal processing framework for learning graphs from two-dimensional signals, modeled as matrix graph signals generated by joint filtering along both dimensions. This formulation leverages the concept of graph stationarity across the two dimensions and leverages product graph representations to capture structured dependencies. Based on this model, we design an optimization problem that can be solved efficiently and provably recovers the optimal underlying Kronecker/Cartesian/strong product graphs. Experiments on synthetic data demonstrate that our approach achieves higher estimation accuracy and reduced computational cost compared to existing methods.
CoVariance Neural Networks (VNNs) perform convolutions on the graph determined by the covariance matrix of the data, which enables expressive and stable covariance-based learning. However, covariance matrices are typically dense, fail to encode conditional independence, and are often precomputed in a task-agnostic way, which may hinder performance. To overcome these limitations, we study Precision Neural Networks (PNNs), i.e., VNNs on the precision matrix—the inverse covariance. The precision matrix naturally encodes statistical independence, often exhibits sparsity, and preserves the covariance spectral structure. To make precision estimation task-aware, we formulate an optimization problem that jointly learns the network parameters and the precision matrix, and solve it via alternating optimization, by sequentially updating the network weights and the precision estimate. We theoretically bound the distance between the estimated and true precision matrices at each iteration, and demonstrate the effectiveness of joint estimation compared to two-step approaches on synthetic and real-world data.
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CoVariance Neural Networks (VNNs) perform convolutions on the graph determined by the covariance matrix of the data, which enables expressive and stable covariance-based learning. However, covariance matrices are typically dense, fail to encode conditional independence, and are often precomputed in a task-agnostic way, which may hinder performance. To overcome these limitations, we study Precision Neural Networks (PNNs), i.e., VNNs on the precision matrix—the inverse covariance. The precision matrix naturally encodes statistical independence, often exhibits sparsity, and preserves the covariance spectral structure. To make precision estimation task-aware, we formulate an optimization problem that jointly learns the network parameters and the precision matrix, and solve it via alternating optimization, by sequentially updating the network weights and the precision estimate. We theoretically bound the distance between the estimated and true precision matrices at each iteration, and demonstrate the effectiveness of joint estimation compared to two-step approaches on synthetic and real-world data.
Journal article
(2026)
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Andrea Cavallo, Ayushman Raghuvanshi, Sundeep Prabhakar Chepuri, Elvin Isufi
Machine learning and data processing techniques relying on covariance information are widespread as they identify meaningful patterns in unsupervised and unlabeled settings. As a prominent example, Principal Component Analysis (PCA) projects data points onto the eigenvectors of their covariance matrix, capturing the directions of maximum variance. This mapping, however, falls short in two directions: it fails to capture information in low-variance directions, relevant when, e.g., the data contains high-variance noise; and it provides unstable results in low-sample regimes, especially when covariance eigenvalues are close. CoVariance Neural Networks (VNNs), i.e., graph neural networks using the covariance matrix as a graph, show improved stability to estimation errors and learn more expressive functions in the covariance spectrum than PCA, but require training and operate in a labeled setup. To get the benefits of both worlds, we propose Covariance Scattering Transforms (CSTs), deep untrained networks that sequentially apply filters localized in the covariance spectrum to the input data and produce expressive hierarchical representations via nonlinearities. We define the filters as covariance wavelets that capture specific and detailed covariance spectral patterns. We improve CSTs’ computational and memory efficiency via a pruning mechanism, and we prove that their error due to finite-sample covariance estimations is less sensitive to close covariance eigenvalues compared to PCA, improving their stability. Our experiments on age prediction from cortical thickness measurements on 4 datasets collecting patients with neurodegenerative diseases show that CSTs produce stable representations in low-data settings, as VNNs but without any training, and lead to comparable or better predictions w.r.t. more complex learning models.
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Machine learning and data processing techniques relying on covariance information are widespread as they identify meaningful patterns in unsupervised and unlabeled settings. As a prominent example, Principal Component Analysis (PCA) projects data points onto the eigenvectors of their covariance matrix, capturing the directions of maximum variance. This mapping, however, falls short in two directions: it fails to capture information in low-variance directions, relevant when, e.g., the data contains high-variance noise; and it provides unstable results in low-sample regimes, especially when covariance eigenvalues are close. CoVariance Neural Networks (VNNs), i.e., graph neural networks using the covariance matrix as a graph, show improved stability to estimation errors and learn more expressive functions in the covariance spectrum than PCA, but require training and operate in a labeled setup. To get the benefits of both worlds, we propose Covariance Scattering Transforms (CSTs), deep untrained networks that sequentially apply filters localized in the covariance spectrum to the input data and produce expressive hierarchical representations via nonlinearities. We define the filters as covariance wavelets that capture specific and detailed covariance spectral patterns. We improve CSTs’ computational and memory efficiency via a pruning mechanism, and we prove that their error due to finite-sample covariance estimations is less sensitive to close covariance eigenvalues compared to PCA, improving their stability. Our experiments on age prediction from cortical thickness measurements on 4 datasets collecting patients with neurodegenerative diseases show that CSTs produce stable representations in low-data settings, as VNNs but without any training, and lead to comparable or better predictions w.r.t. more complex learning models.
Covariance Neural Networks (VNNs) perform graph convolutions on the empirical covariance matrix of signals defined over finite-dimensional Hilbert spaces, motivated by robustness and transferability properties. Yet, little is known about how these arguments extend to infinite-dimensional Hilbert spaces. In this work, we take a first step by introducing a novel convolutional learning framework for signals defined over infinite-dimensional Hilbert spaces, centered on the (empirical) covariance operator. We constructively define Hilbert coVariance Filters (HVFs) and design Hilbert coVariance Networks (HVNs) as stacks of HVF filterbanks with nonlinear activations. We propose a principled discretization procedure, and we prove that empirical HVFs can recover the Functional PCA (FPCA) of the filtered signals. We then describe the versatility of our framework with examples ranging from multivariate real-valued functions to reproducing kernel Hilbert spaces. Finally, we validate HVNs on both synthetic and real-world time-series classification tasks, showing robust performance compared to MLP and FPCA-based classifiers.
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Covariance Neural Networks (VNNs) perform graph convolutions on the empirical covariance matrix of signals defined over finite-dimensional Hilbert spaces, motivated by robustness and transferability properties. Yet, little is known about how these arguments extend to infinite-dimensional Hilbert spaces. In this work, we take a first step by introducing a novel convolutional learning framework for signals defined over infinite-dimensional Hilbert spaces, centered on the (empirical) covariance operator. We constructively define Hilbert coVariance Filters (HVFs) and design Hilbert coVariance Networks (HVNs) as stacks of HVF filterbanks with nonlinear activations. We propose a principled discretization procedure, and we prove that empirical HVFs can recover the Functional PCA (FPCA) of the filtered signals. We then describe the versatility of our framework with examples ranging from multivariate real-valued functions to reproducing kernel Hilbert spaces. Finally, we validate HVNs on both synthetic and real-world time-series classification tasks, showing robust performance compared to MLP and FPCA-based classifiers.
Topological spaces capture richer relationships than graphs by modeling interactions not only between nodes but also among higher-order entities, such as edges or triangles. This motivates the representation of information defined in irregular domains as topological signals.We focus on simplicial complexes, which are collections of simplices (nodes, edges, triangles and so on) adhering to the inclusion property: a higher-order element cannot exist unless its lower-order components are also part of the complex (e.g., a triangle requires its edges). Accordingly, simplicial signals are functions defined over this set of simplices. By leveraging the spectral dualities of Hodge and Dirac theory, practical simplicial signals often concentrate in specific spectral subspaces (e.g., gradient or curl). For instance, in a foreign currency exchange network, the exchange flow signals typically satisfy the arbitrage-free condition and hence are curl-free. However, the presence of anomalies can disrupt these conditions, causing the signals to deviate from such subspaces. In this work, we formulate a hypothesis testing framework to detect whether simplicial complex signals lie in specific subspaces in a principled and tractable manner. Concretely, we propose Neyman-Pearson matched topological subspace detectors for signals defined at a single simplicial level (such as edges) or jointly across all levels of a simplicial complex. The (energy-based projection) proposed detectors handle missing values, provide closed-form performance analysis, and effectively capture the unique topo-logical properties of the data. We demonstrate the effectiveness of the proposed topological detectors on various real-world data, including foreign currency exchange networks.
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Topological spaces capture richer relationships than graphs by modeling interactions not only between nodes but also among higher-order entities, such as edges or triangles. This motivates the representation of information defined in irregular domains as topological signals.We focus on simplicial complexes, which are collections of simplices (nodes, edges, triangles and so on) adhering to the inclusion property: a higher-order element cannot exist unless its lower-order components are also part of the complex (e.g., a triangle requires its edges). Accordingly, simplicial signals are functions defined over this set of simplices. By leveraging the spectral dualities of Hodge and Dirac theory, practical simplicial signals often concentrate in specific spectral subspaces (e.g., gradient or curl). For instance, in a foreign currency exchange network, the exchange flow signals typically satisfy the arbitrage-free condition and hence are curl-free. However, the presence of anomalies can disrupt these conditions, causing the signals to deviate from such subspaces. In this work, we formulate a hypothesis testing framework to detect whether simplicial complex signals lie in specific subspaces in a principled and tractable manner. Concretely, we propose Neyman-Pearson matched topological subspace detectors for signals defined at a single simplicial level (such as edges) or jointly across all levels of a simplicial complex. The (energy-based projection) proposed detectors handle missing values, provide closed-form performance analysis, and effectively capture the unique topo-logical properties of the data. We demonstrate the effectiveness of the proposed topological detectors on various real-world data, including foreign currency exchange networks.
Journal article
(2026)
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Roberto Bentivoglio, Sebastiaan Nicolas Jonkman, Elvin Isufi, Riccardo Taormina
Flood hazard maps are essential for protection and emergency plans, yet their probabilistic application is constrained by the computational cost of numerical models. Deep learning surrogates can provide orders of magnitude faster predictions, but their use for uncertainty quantification in realistic settings and their ability to incorporate hydraulic structures remain largely unexplored. Studying deep learning surrogates for probabilistic flood maps is non-trivial because of the lack of reference ground-truth data that might lead to misleading confidence in predictions. Moreover, hydraulic structures are challenging to include due to their generally unidimensional nature. In this work, we investigate the use of deep learning surrogates for realistic, large-scale flood simulations in case studies with hydraulic structures under diverse boundary conditions. To this end, we employ the multi-scale shallow-water-equations graph neural network (mSWE-GNN) that enjoys transferability to different boundary conditions and locations and whose graph-based architecture allows to represent structures such as canals, underpasses, and elevated elements as inputs. To address the lack of reference ground-truth data, we further introduce the average relative mass error (ARME), a mass-conservation-based criterion that helps identify physically plausible simulations. We applied the model on dike ring 41 in the Netherlands, generating probabilistic flood maps that account for uncertainties in breach location and breach outflow hydrographs. The model was trained on 30 simulations, generated with Delft3D, and evaluated against unseen benchmark simulations from the Dutch national flood catalogue, achieving a critical success index (CSI) of 73.6 % while running 10 000 times faster than the numerical simulator. The proposed ARME is negatively correlated with the CSI, with a Spearman correlation coefficient of -0.7, making it a useful indicator of simulation plausibility when evaluating unseen case studies. We obtained probabilistic flood maps by running 10 000 different flooding scenarios on a computational mesh of 180 000 cells in approximately 10 h, with about half of the simulations classified as plausible based on the mass-conservation check. This framework offers a practical tool for rapid probabilistic flood hazard assessment and a way to prioritize detailed physical simulations, supporting more efficient and robust flood risk management.
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Flood hazard maps are essential for protection and emergency plans, yet their probabilistic application is constrained by the computational cost of numerical models. Deep learning surrogates can provide orders of magnitude faster predictions, but their use for uncertainty quantification in realistic settings and their ability to incorporate hydraulic structures remain largely unexplored. Studying deep learning surrogates for probabilistic flood maps is non-trivial because of the lack of reference ground-truth data that might lead to misleading confidence in predictions. Moreover, hydraulic structures are challenging to include due to their generally unidimensional nature. In this work, we investigate the use of deep learning surrogates for realistic, large-scale flood simulations in case studies with hydraulic structures under diverse boundary conditions. To this end, we employ the multi-scale shallow-water-equations graph neural network (mSWE-GNN) that enjoys transferability to different boundary conditions and locations and whose graph-based architecture allows to represent structures such as canals, underpasses, and elevated elements as inputs. To address the lack of reference ground-truth data, we further introduce the average relative mass error (ARME), a mass-conservation-based criterion that helps identify physically plausible simulations. We applied the model on dike ring 41 in the Netherlands, generating probabilistic flood maps that account for uncertainties in breach location and breach outflow hydrographs. The model was trained on 30 simulations, generated with Delft3D, and evaluated against unseen benchmark simulations from the Dutch national flood catalogue, achieving a critical success index (CSI) of 73.6 % while running 10 000 times faster than the numerical simulator. The proposed ARME is negatively correlated with the CSI, with a Spearman correlation coefficient of -0.7, making it a useful indicator of simulation plausibility when evaluating unseen case studies. We obtained probabilistic flood maps by running 10 000 different flooding scenarios on a computational mesh of 180 000 cells in approximately 10 h, with about half of the simulations classified as plausible based on the mass-conservation check. This framework offers a practical tool for rapid probabilistic flood hazard assessment and a way to prioritize detailed physical simulations, supporting more efficient and robust flood risk management.
We study the problem of generating graph signals from unknown distributions defined over given graphs, relevant to domains such as recommender systems or sensor networks. Our approach builds on generative diffusion models, which are well established in vision and graph generation but remain underexplored for graph signals. Existing methods lack generality, either ignoring the graph structure in the forward process or designing graph-aware mechanisms tailored to specific domains. We adopt a forward process that incorporates the graph through the heat equation. Rather than relying on the standard formulation, we consider a time-warped coefficient to mitigate the exponential decay of the drift term, yielding a graph-aware generative diffusion model (GAD). We analyze its forward dynamics, proving convergence to a Gaussian Markov random field with covariance parametrized by the graph Laplacian, and interpret the backward dynamics as a sequence of graph-signal denoising problems. Finally, we demonstrate the advantages of GAD on synthetic data, real traffic speed measurements, and a temperature sensor network.
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We study the problem of generating graph signals from unknown distributions defined over given graphs, relevant to domains such as recommender systems or sensor networks. Our approach builds on generative diffusion models, which are well established in vision and graph generation but remain underexplored for graph signals. Existing methods lack generality, either ignoring the graph structure in the forward process or designing graph-aware mechanisms tailored to specific domains. We adopt a forward process that incorporates the graph through the heat equation. Rather than relying on the standard formulation, we consider a time-warped coefficient to mitigate the exponential decay of the drift term, yielding a graph-aware generative diffusion model (GAD). We analyze its forward dynamics, proving convergence to a Gaussian Markov random field with covariance parametrized by the graph Laplacian, and interpret the backward dynamics as a sequence of graph-signal denoising problems. Finally, we demonstrate the advantages of GAD on synthetic data, real traffic speed measurements, and a temperature sensor network.
Learning deep representations from covariance in-formation via coVariance Neural Networks (VNNs) has shown an improved performance and insights with respect to Principal Component Analysis (PCA)-based alternatives and better stability in finite-sample regimes. VNNs extend the PCA transform by learning end-to-end the spectral processing function on the principal directions of the data in each layer. However, VNNs operate on the pre-computed sample covariance matrix, which is prone to estimation errors, sensitive to outliers, and not adapted to the task at hand. To overcome this limitation, we propose Robust coVariance Neural Networks (RVNNs), a framework that simultaneously learns a robust estimator of the covariance matrix and the VNN parameters in an end-to-end manner, leading to a fully task-aware pipeline. We prove that RVNNs combine the robustness to outliers with the finite-sample stability of VNNs, and we show that their end-to-end robust covariance learning leads to better prediction performance compared to robust PCA-based approaches on simulated and real-world data from brain recordings and human motion sensor measurements.
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Learning deep representations from covariance in-formation via coVariance Neural Networks (VNNs) has shown an improved performance and insights with respect to Principal Component Analysis (PCA)-based alternatives and better stability in finite-sample regimes. VNNs extend the PCA transform by learning end-to-end the spectral processing function on the principal directions of the data in each layer. However, VNNs operate on the pre-computed sample covariance matrix, which is prone to estimation errors, sensitive to outliers, and not adapted to the task at hand. To overcome this limitation, we propose Robust coVariance Neural Networks (RVNNs), a framework that simultaneously learns a robust estimator of the covariance matrix and the VNN parameters in an end-to-end manner, leading to a fully task-aware pipeline. We prove that RVNNs combine the robustness to outliers with the finite-sample stability of VNNs, and we show that their end-to-end robust covariance learning leads to better prediction performance compared to robust PCA-based approaches on simulated and real-world data from brain recordings and human motion sensor measurements.
Topological signal processing and learning
Recent advances and future challenges
Journal article
(2025)
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Elvin Isufi, Geert Leus, Baltasar Beferull-Lozano, Sergio Barbarossa, Paolo Di Lorenzo
Developing methods to process irregularly structured data is crucial in applications like gene-regulatory, brain, power, and socioeconomic networks. Graphs have been the go-to algebraic tool for modeling the structure via nodes and edges capturing their interactions, leading to the establishment of the fields of graph signal processing (GSP) and graph machine learning (GML). Key graph-aware methods include Fourier transform, filtering, sampling, as well as topology identification and spatiotemporal processing. Although versatile, graphs can model only pairwise dependencies in the data. To this end, topological structures such as simplicial and cell complexes have emerged as algebraic representations for more intricate structure modeling in data-driven systems, fueling the rapid development of novel topological-based processing and learning methods. This paper first presents the core principles of topological signal processing through the Hodge theory, a framework instrumental in propelling the field forward thanks to principled connections with GSP-GML. It then outlines advances in topological signal representation, filtering, and sampling, as well as inferring topological structures from data, processing spatiotemporal topological signals, and connections with topological machine learning. The impact of topological signal processing and learning is finally highlighted in applications dealing with flow data over networks, geometric processing, statistical ranking, biology, and semantic communication.
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Developing methods to process irregularly structured data is crucial in applications like gene-regulatory, brain, power, and socioeconomic networks. Graphs have been the go-to algebraic tool for modeling the structure via nodes and edges capturing their interactions, leading to the establishment of the fields of graph signal processing (GSP) and graph machine learning (GML). Key graph-aware methods include Fourier transform, filtering, sampling, as well as topology identification and spatiotemporal processing. Although versatile, graphs can model only pairwise dependencies in the data. To this end, topological structures such as simplicial and cell complexes have emerged as algebraic representations for more intricate structure modeling in data-driven systems, fueling the rapid development of novel topological-based processing and learning methods. This paper first presents the core principles of topological signal processing through the Hodge theory, a framework instrumental in propelling the field forward thanks to principled connections with GSP-GML. It then outlines advances in topological signal representation, filtering, and sampling, as well as inferring topological structures from data, processing spatiotemporal topological signals, and connections with topological machine learning. The impact of topological signal processing and learning is finally highlighted in applications dealing with flow data over networks, geometric processing, statistical ranking, biology, and semantic communication.
In urban centers, cycling is increasingly popular as an eco-friendly transportation mode and a short-distance transport option, driving higher demand for accurate bicycle travel time estimation. Policymakers need to understand bicycle traffic for urban traffic management and sustainable transport promotion, while cyclists benefit from better route planning and improved network efficiency. However, urban bicycle travel time estimation has not received as much attention as car traffic estimation and presents several challenges: 1) Limited availability of structural cycling data, which can be inaccessible due to privacy concerns and/or severely biased by user demographics. 2) The diverse and complex behaviors of cyclists. 3) The lack of strict road constraints for cyclists and frequent rule violations, complicating the model definition of a comprehensive cycling infrastructure network. This paper presents the first study on urban bicycle travel time estimation using GPS tracking data. Leveraging graph-based deep learning's ability to learn from topological network information, we introduce the Dual Graph-based approach for bicycles (DG4b), which employs two parallel encode-process-decode pipelines: one for a shared undirected road network graph to capture intrinsic road characteristics, and another for a directed trip-specific graph reflecting unique trip features. The outputs are combined to estimate road segment speeds and overall trip travel time. When applied to a real-world dataset from Berlin, our method shows superior accuracy and reliability compared to baseline models, while maintaining low complexity. Our approach provides a novel perspective on integrating bicycling-specific characteristics and aims to inspire more future research in bicycle-related traffic estimation.
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In urban centers, cycling is increasingly popular as an eco-friendly transportation mode and a short-distance transport option, driving higher demand for accurate bicycle travel time estimation. Policymakers need to understand bicycle traffic for urban traffic management and sustainable transport promotion, while cyclists benefit from better route planning and improved network efficiency. However, urban bicycle travel time estimation has not received as much attention as car traffic estimation and presents several challenges: 1) Limited availability of structural cycling data, which can be inaccessible due to privacy concerns and/or severely biased by user demographics. 2) The diverse and complex behaviors of cyclists. 3) The lack of strict road constraints for cyclists and frequent rule violations, complicating the model definition of a comprehensive cycling infrastructure network. This paper presents the first study on urban bicycle travel time estimation using GPS tracking data. Leveraging graph-based deep learning's ability to learn from topological network information, we introduce the Dual Graph-based approach for bicycles (DG4b), which employs two parallel encode-process-decode pipelines: one for a shared undirected road network graph to capture intrinsic road characteristics, and another for a directed trip-specific graph reflecting unique trip features. The outputs are combined to estimate road segment speeds and overall trip travel time. When applied to a real-world dataset from Berlin, our method shows superior accuracy and reliability compared to baseline models, while maintaining low complexity. Our approach provides a novel perspective on integrating bicycling-specific characteristics and aims to inspire more future research in bicycle-related traffic estimation.
Neural networks on simplicial complexes (SCs) can learn representations from data residing on simplices such as nodes, edges, triangles, etc. However, existing works often overlook the Hodge theorem that decomposes simplicial data into three orthogonal characteristic subspaces, such as the identifiable gradient, curl and harmonic components of edge flows. This provides a universal tool to understand the machine learning models on SCs, thus, allowing for better principled and effective learning. In this paper, we study the effect of this data inductive bias on learning on SCs via the principle of convolutions. Particularly, we present a general convolutional architecture that respects the three key principles of uncoupling the lower and upper simplicial adjacencies, accounting for the inter-simplicial couplings, and performing higher-order convolutions. To understand these principles, we first use Dirichlet energy minimizations on SCs to interpret their effects on mitigating simplicial oversmoothing. Then, we show the three principles promote the Hodge-aware learning of this architecture, through the lens of spectral simplicial theory, in the sense that the three Hodge subspaces are invariant under its learnable functions and the learning in two nontrivial subspaces is independent and expressive. Third, we investigate the learning ability of this architecture in optic of perturbation theory on simplicial topologies and prove that the convolutional architecture is stable to small perturbations. Finally, we corroborate the three principles by comparing with methods that either violate or do not respect them. Overall, this paper bridges learning on SCs with the Hodge theorem, highlighting its importance for rational and effective learning from simplicial data, and provides theoretical insights to convolutional learning on SCs.
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Neural networks on simplicial complexes (SCs) can learn representations from data residing on simplices such as nodes, edges, triangles, etc. However, existing works often overlook the Hodge theorem that decomposes simplicial data into three orthogonal characteristic subspaces, such as the identifiable gradient, curl and harmonic components of edge flows. This provides a universal tool to understand the machine learning models on SCs, thus, allowing for better principled and effective learning. In this paper, we study the effect of this data inductive bias on learning on SCs via the principle of convolutions. Particularly, we present a general convolutional architecture that respects the three key principles of uncoupling the lower and upper simplicial adjacencies, accounting for the inter-simplicial couplings, and performing higher-order convolutions. To understand these principles, we first use Dirichlet energy minimizations on SCs to interpret their effects on mitigating simplicial oversmoothing. Then, we show the three principles promote the Hodge-aware learning of this architecture, through the lens of spectral simplicial theory, in the sense that the three Hodge subspaces are invariant under its learnable functions and the learning in two nontrivial subspaces is independent and expressive. Third, we investigate the learning ability of this architecture in optic of perturbation theory on simplicial topologies and prove that the convolutional architecture is stable to small perturbations. Finally, we corroborate the three principles by comparing with methods that either violate or do not respect them. Overall, this paper bridges learning on SCs with the Hodge theorem, highlighting its importance for rational and effective learning from simplicial data, and provides theoretical insights to convolutional learning on SCs.
We propose a topology-aware online framework for detecting and localizing change points in partially observed temporal signals defined over cellular complexes. We model the process as a linear state space model. The latent dynamics follow a topology-consistent stochastic partial differential equation (SPDE), and the observations a topologically-filtered version of the state. Hidden states are inferred via a Kalman filter, while model parameters are adapted online through likelihood-based updates. Change points are detected by tracking deviations in the uncertainty parameters using a score-based criterion, and the responsible components are localized by mapping these deviations to specific cells in the complex. Under missing or noisy observations, backward Kalman smoothing—reinitialized at each detected change—provides consistent state reconstruction and reliable imputation. Experiments on synthetic and EPANET-simulated water networks demonstrate accurate detection, precise localization, and robust reconstruction under partial observability.
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We propose a topology-aware online framework for detecting and localizing change points in partially observed temporal signals defined over cellular complexes. We model the process as a linear state space model. The latent dynamics follow a topology-consistent stochastic partial differential equation (SPDE), and the observations a topologically-filtered version of the state. Hidden states are inferred via a Kalman filter, while model parameters are adapted online through likelihood-based updates. Change points are detected by tracking deviations in the uncertainty parameters using a score-based criterion, and the responsible components are localized by mapping these deviations to specific cells in the complex. Under missing or noisy observations, backward Kalman smoothing—reinitialized at each detected change—provides consistent state reconstruction and reliable imputation. Experiments on synthetic and EPANET-simulated water networks demonstrate accurate detection, precise localization, and robust reconstruction under partial observability.
Inferring higher-order network structures from nodal data is an emerging challenge across fields such as signal processing, machine learning, and causal inference. While directed acyclic graphs (DAGs) provide a powerful framework for modeling causal or functional dependencies, they only capture pairwise interactions. This paper introduces a new directed acyclic hypergraph (DAH) signal model that generalizes DAG-based structural equation modeling to multi-node (higher-order) relationships. Our approach begins by lifting a directed hyper-graph (DH) into an equivalent bipartite directed graph (DG), where virtual nodes represent source-node sets of hyperedges. Nodal data are assigned to both original and virtual nodes, and an SEM is defined over the DG. By imposing a smooth acyclicity constraint on this bipartite graph, we obtain a continuous and scalable formulation for DAH estimation from nodal observations. The proposed framework unifies hypergraph learning and DAG inference under a common optimization perspective, enabling interpretable higher-order dependency discovery. Numerical experiments demonstrate the ability of the method to recover meaningful DAH structures from simulated nodal data.
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Inferring higher-order network structures from nodal data is an emerging challenge across fields such as signal processing, machine learning, and causal inference. While directed acyclic graphs (DAGs) provide a powerful framework for modeling causal or functional dependencies, they only capture pairwise interactions. This paper introduces a new directed acyclic hypergraph (DAH) signal model that generalizes DAG-based structural equation modeling to multi-node (higher-order) relationships. Our approach begins by lifting a directed hyper-graph (DH) into an equivalent bipartite directed graph (DG), where virtual nodes represent source-node sets of hyperedges. Nodal data are assigned to both original and virtual nodes, and an SEM is defined over the DG. By imposing a smooth acyclicity constraint on this bipartite graph, we obtain a continuous and scalable formulation for DAH estimation from nodal observations. The proposed framework unifies hypergraph learning and DAG inference under a common optimization perspective, enabling interpretable higher-order dependency discovery. Numerical experiments demonstrate the ability of the method to recover meaningful DAH structures from simulated nodal data.
Topological Deep Learning (TDL) has emerged as a paradigm to process and learn from signals defined on higher-order combinatorial topological spaces, such as simplicial or cell complexes. Although many complex systems have an asymmetric relational structure, most TDL models forcibly symmetrize these relationships. In this paper, we first introduce a novel notion of higher-order directionality and we then design Directed Simplicial Neural Networks (Dir-SNNs) based on it. Dir-SNNs are message-passing networks operating on directed simplicial complexes able to leverage directed and possibly asymmetric interactions among the simplices. To our knowledge, this is the first TDL model using a notion of higher-order directionality. We theoretically and empirically prove that Dir-SNNs are more expressive than their directed graph counterpart in distinguishing non-isomorphic directed graphs. Experiments on a synthetic source localization task demonstrate that Dir-SNNs outperform undirected SNNs when the underlying complex is directed, and perform comparably when the underlying complex is undirected.
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Topological Deep Learning (TDL) has emerged as a paradigm to process and learn from signals defined on higher-order combinatorial topological spaces, such as simplicial or cell complexes. Although many complex systems have an asymmetric relational structure, most TDL models forcibly symmetrize these relationships. In this paper, we first introduce a novel notion of higher-order directionality and we then design Directed Simplicial Neural Networks (Dir-SNNs) based on it. Dir-SNNs are message-passing networks operating on directed simplicial complexes able to leverage directed and possibly asymmetric interactions among the simplices. To our knowledge, this is the first TDL model using a notion of higher-order directionality. We theoretically and empirically prove that Dir-SNNs are more expressive than their directed graph counterpart in distinguishing non-isomorphic directed graphs. Experiments on a synthetic source localization task demonstrate that Dir-SNNs outperform undirected SNNs when the underlying complex is directed, and perform comparably when the underlying complex is undirected.
Journal article
(2025)
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Roberto Bentivoglio, Elvin Isufi, Sebastiaan Nicolas Jonkman, Riccardo Taormina
Deep-learning-based surrogate models represent a powerful alternative to numerical models for speeding up flood mapping while preserving accuracy. In particular, solutions based on hydraulic-based graph neural networks (SWE-GNNs) enable transferability to domains not used for training and allow the inclusion of physical constraints. However, these models are limited due to four main aspects. First, they cannot model rapid differences in flow propagation speeds; secondly, they can face instabilities during training when using a large number of layers, needed for effective modelling; third, they cannot accommodate time-varying boundary conditions; and fourth, they require initial conditions from a numerical solver. To address these issues, we propose a multi-scale hydraulic-based graph neural network (mSWE-GNN) that models the flood at different resolutions and propagation speeds. We include time-varying boundary conditions via ghost cells, which enforce the solution at the domain’s boundary and drop the need for a numerical solver for the initial conditions. To improve generalization over unseen meshes and reduce the data demand, we use invariance principles and make the inputs independent from coordinates' rotations. Numerical results applied to dike-breach floods show that the model predicts the full spatio-temporal simulation of the flood over unseen irregular meshes, topographies, and time-varying boundary conditions, with mean absolute errors in time of 0.05 m for water depths and 0.003 m2 s−1 for unit discharges. We further corroborate the mSWE-GNN in a realistic case study in the Netherlands and show generalization capabilities with only one fine-tuning sample, with mean absolute errors of 0.12 m for water depth, a critical success index for a water depth threshold of 0.05 m of 87.68 %, and speed-ups of over 700 times. Overall, the approach opens up several avenues for probabilistic analyses of realistic configurations and flood scenarios.
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Deep-learning-based surrogate models represent a powerful alternative to numerical models for speeding up flood mapping while preserving accuracy. In particular, solutions based on hydraulic-based graph neural networks (SWE-GNNs) enable transferability to domains not used for training and allow the inclusion of physical constraints. However, these models are limited due to four main aspects. First, they cannot model rapid differences in flow propagation speeds; secondly, they can face instabilities during training when using a large number of layers, needed for effective modelling; third, they cannot accommodate time-varying boundary conditions; and fourth, they require initial conditions from a numerical solver. To address these issues, we propose a multi-scale hydraulic-based graph neural network (mSWE-GNN) that models the flood at different resolutions and propagation speeds. We include time-varying boundary conditions via ghost cells, which enforce the solution at the domain’s boundary and drop the need for a numerical solver for the initial conditions. To improve generalization over unseen meshes and reduce the data demand, we use invariance principles and make the inputs independent from coordinates' rotations. Numerical results applied to dike-breach floods show that the model predicts the full spatio-temporal simulation of the flood over unseen irregular meshes, topographies, and time-varying boundary conditions, with mean absolute errors in time of 0.05 m for water depths and 0.003 m2 s−1 for unit discharges. We further corroborate the mSWE-GNN in a realistic case study in the Netherlands and show generalization capabilities with only one fine-tuning sample, with mean absolute errors of 0.12 m for water depth, a critical success index for a water depth threshold of 0.05 m of 87.68 %, and speed-ups of over 700 times. Overall, the approach opens up several avenues for probabilistic analyses of realistic configurations and flood scenarios.
This paper proposes a scalable method for identifying interactions in higher-order networks from observations of nodal processes. Finding such dependencies is important in many disciplines, including neuroscience, social influence modeling, and beyond. However, current approaches are either limited to extracting pairwise dependencies or struggle with scalability, as estimating higher-order dependencies becomes computationally prohibitive. To overcome these challenges, we introduce a tensorbased graph Volterra model that leverages low-rank decomposition techniques to estimate higher-order interactions efficiently. Our approach not only reduces computational and storage complexity but also acts as an implicit regularizer, improving network estimation in ill-posed settings. We validate our method through simulations and real data experiments, demonstrating competitive performance and enhanced scalability compared to existing techniques.
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This paper proposes a scalable method for identifying interactions in higher-order networks from observations of nodal processes. Finding such dependencies is important in many disciplines, including neuroscience, social influence modeling, and beyond. However, current approaches are either limited to extracting pairwise dependencies or struggle with scalability, as estimating higher-order dependencies becomes computationally prohibitive. To overcome these challenges, we introduce a tensorbased graph Volterra model that leverages low-rank decomposition techniques to estimate higher-order interactions efficiently. Our approach not only reduces computational and storage complexity but also acts as an implicit regularizer, improving network estimation in ill-posed settings. We validate our method through simulations and real data experiments, demonstrating competitive performance and enhanced scalability compared to existing techniques.
Temporal networks arise due to certain dynamics influencing their connections or due to the change in interactions between the nodes themselves, as seen for example in social networks. Such evolution can be algebraically represented by a three-way tensor, which lends itself to using tensor decompositions to study the underpinning factors driving the network evolution. Low rank tensor decompositions have been used for temporal networks but mostly with a focus on downstream tasks and have been seldom used to study the temporal network itself. Here, we use the tensor decomposition to identify a limited number of key mode graphs that can explain the temporal network, and which linear combination can represent its evolution. For this, we put for a novel graph-based tensor decomposition approach where we impose a graph structure on the two modes of the tensor and a smoothness on the temporal dimension. We use these mode graphs to investigate the temporal network and corroborate their usability for network reconstruction and link prediction.
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Temporal networks arise due to certain dynamics influencing their connections or due to the change in interactions between the nodes themselves, as seen for example in social networks. Such evolution can be algebraically represented by a three-way tensor, which lends itself to using tensor decompositions to study the underpinning factors driving the network evolution. Low rank tensor decompositions have been used for temporal networks but mostly with a focus on downstream tasks and have been seldom used to study the temporal network itself. Here, we use the tensor decomposition to identify a limited number of key mode graphs that can explain the temporal network, and which linear combination can represent its evolution. For this, we put for a novel graph-based tensor decomposition approach where we impose a graph structure on the two modes of the tensor and a smoothness on the temporal dimension. We use these mode graphs to investigate the temporal network and corroborate their usability for network reconstruction and link prediction.
Graph filters are a staple tool for processing signals over graphs in a multitude of downstream tasks. However, they are commonly designed for graphs with a fixed number of nodes, despite real-world networks typically grow over time. This topological evolution is often known up to a stochastic model, thus, making conventional graph filters ill-equipped to withstand such topological changes, their uncertainty, as well as the dynamic nature of the incoming data. To tackle these issues, we propose an online graph filtering framework by relying on online learning principles. We design filters for scenarios where the topology is both known and unknown, including a learner adaptive to such evolution. We conduct a regret analysis to highlight the role played by the different components such as the online algorithm, the filter order, and the growing graph model. Numerical experiments with synthetic and real data corroborate the proposed approach for graph signal inference tasks and show a competitive performance w.r.t. baselines and state-of-the-art alternatives.
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Graph filters are a staple tool for processing signals over graphs in a multitude of downstream tasks. However, they are commonly designed for graphs with a fixed number of nodes, despite real-world networks typically grow over time. This topological evolution is often known up to a stochastic model, thus, making conventional graph filters ill-equipped to withstand such topological changes, their uncertainty, as well as the dynamic nature of the incoming data. To tackle these issues, we propose an online graph filtering framework by relying on online learning principles. We design filters for scenarios where the topology is both known and unknown, including a learner adaptive to such evolution. We conduct a regret analysis to highlight the role played by the different components such as the online algorithm, the filter order, and the growing graph model. Numerical experiments with synthetic and real data corroborate the proposed approach for graph signal inference tasks and show a competitive performance w.r.t. baselines and state-of-the-art alternatives.
Inference of time varying data over graphs is of importance in real-world applications such as urban water networks, economics, and brain recordings. It typically relies on identifying a computationally affordable joint spatiotemporal method that can leverage the patterns in the data. While this per se is a challenging task, it becomes even more so when the network comes with uncertainties, which, if not accounted for, can lead to unpredictable consequences. To target this setting, we model graph uncertainties as Gaussian noise on the edges and design a stochastic partial differential equation (SPDE) based on it. We use this SPDE as a state equation to model the time varying signal evolution and extend it further to a state-space model where the observations are graph-filtered versions of the state. This allows us to have a joint spatiotemporal expressive kernel that can be estimated online via Kalman filtering and which parameters can also be estimated online via maximum likelihood principles, ultimately, reducing the computational cost. We corroborate the proposed approach on numerical experiments, showing a superior performance to approaches ignoring either the uncertainty or considering a separable spatiotemporal kernel.
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Inference of time varying data over graphs is of importance in real-world applications such as urban water networks, economics, and brain recordings. It typically relies on identifying a computationally affordable joint spatiotemporal method that can leverage the patterns in the data. While this per se is a challenging task, it becomes even more so when the network comes with uncertainties, which, if not accounted for, can lead to unpredictable consequences. To target this setting, we model graph uncertainties as Gaussian noise on the edges and design a stochastic partial differential equation (SPDE) based on it. We use this SPDE as a state equation to model the time varying signal evolution and extend it further to a state-space model where the observations are graph-filtered versions of the state. This allows us to have a joint spatiotemporal expressive kernel that can be estimated online via Kalman filtering and which parameters can also be estimated online via maximum likelihood principles, ultimately, reducing the computational cost. We corroborate the proposed approach on numerical experiments, showing a superior performance to approaches ignoring either the uncertainty or considering a separable spatiotemporal kernel.