Samuel Rey

Hamed Ajorlou, Samuel Rey, Gonzalo Mateos et al.

Learning the structure of directed acyclic graphs (DAGs) from observational data is a central problem in causal discovery, statistical signal processing, and machine learning. Under a linear Gaussian structural equation model (SEM) with equal noise variances, the problem is identifiable and we show that the ensemble precision matrix of the observations exhibits a distinctive structure that facilitates DAG recovery. Exploiting this property, we propose BUILD (Bottom-Up Inference of Linear DAGs), a deterministic stepwise algorithm that identifies leaf nodes and their parents, then prunes the leaves by removing incident edges to proceed to the next step, exactly reconstructing the DAG from the true precision matrix. In practice, precision matrices must be estimated from finite data, and ill-conditioning may lead to error accumulation across BUILD steps. As a mitigation strategy, we periodically re-estimate the precision matrix (with less variables as leaves are pruned), trading off runtime for enhanced robustness. Reproducible results on challenging synthetic benchmarks demonstrate that BUILD compares favorably to state-of-the-art DAG learning algorithms, while offering an explicit handle on complexity.

Samuel Rey, Madeline Navarro, Andrei Buciulea et al.

Graph learning problems are typically approached by focusing on learning the topology of a single graph when signals from all nodes are available. However, many contemporary setups involve multiple related networks and, moreover, it is often the case that only a subset of nodes is observed while the rest remain hidden. Motivated by this, we propose a joint graph learning method that takes into account the presence of hidden (latent) variables. Intuitively, the presence of the hidden nodes renders the inference task ill-posed and challenging to solve, so we overcome this detrimental influence by harnessing the similarity of the estimated graphs. To that end, we assume that the observed signals are drawn from a Gaussian Markov random field with latent variables and we carefully model the graph similarity among hidden (latent) nodes. Then, we exploit the structure resulting from the previous considerations to propose a convex optimization problem that solves the joint graph learning task by providing a regularized maximum likelihood estimator. Finally, we compare the proposed algorithm with different baselines and evaluate its performance over synthetic and real-world graphs.

Samuel Rey, Bishwadeep Das, Elvin Isufi

This paper addresses the problem of online network topology inference for expanding graphs from a stream of spatiotemporal signals. Online algorithms for dynamic graph learning are crucial in delay-sensitive applications or when changes in topology occur rapidly. While existing works focus on inferring the connectivity within a fixed set of nodes, in practice, the graph can grow as new nodes join the network. This poses additional challenges like modeling temporal dynamics involving signals and graphs of different sizes. This growth also increases the computational complexity of the learning process, which may become prohibitive. To the best of our knowledge, this is the first work to tackle this setting. We propose a general online algorithm based on projected proximal gradient descent that accounts for the increasing graph size at each iteration. Recursively updating the sample covariance matrix is a key aspect of our approach. We introduce a strategy that enables different types of updates for nodes that just joined the network and for previously existing nodes. To provide further insights into the proposed method, we specialize it in Gaussian Markov random field settings, where we analyze the computational complexity and characterize the dynamic cumulative regret. Finally, we demonstrate the effectiveness of the proposed approach using both controlled experiments and real-world datasets from epidemic and financial networks.

Samuel Rey, Madeline Navarro, Victor M. Tenorio et al.

Graph neural networks (GNNs) have become a workhorse approach for learning from data defined over irregular domains, typically by implicitly assuming that the data structure is represented by a homophilic graph. However, recent works have revealed that many relevant applications involve heterophilic data where the performance of GNNs can be notably compromised. To address this challenge, we present a simple yet effective architecture designed to mitigate the limitations of the homophily assumption. The proposed architecture reinterprets the role of graph filters in convolutional GNNs, resulting in a more general architecture while incorporating a stronger inductive bias than GNNs based on filter banks. The proposed convolutional layer enhances the expressive capacity of the architecture enabling it to learn from both homophilic and heterophilic data and preventing the issue of oversmoothing. From a theoretical standpoint, we show that the proposed architecture is permutation equivariant. Finally, we show that the proposed GNNs compares favorably relative to several state-of-the-art baselines in both homophilic and heterophilic datasets, showcasing its promising potential.

Samuel Rey, Madeline navarro, Gonzalo Mateos

This work addresses the problem of learning directed acyclic graphs (DAGs) from nodal observations generated by a linear structural equation model. DAG learning is a central task in signal processing, machine learning, and causal inference, but it remains challenging because acyclicity is a global combinatorial property. Continuous acyclicity constraints have led to important algorithmic advances by replacing the discrete DAG constraint with smooth equality constraints. However, existing formulations still involve difficult non-convex optimization landscapes and may suffer from degenerate first-order optimality conditions. Here, we restrict attention to DAGs with non-negative edge weights and exploit this additional structure to obtain a simpler characterization of acyclicity. Building on this characterization, we formulate a regularized non-negative DAG learning problem and develop an algorithm based on the method of multipliers. We further analyze the benign optimization landscape induced by non-negativity. In the population regime, we show that the true DAG is the unique global minimizer of the proposed augmented-Lagrangian formulation; moreover, the landscape contains no spurious interior stationary points, and the true DAG is the only acyclic KKT point. Numerical experiments on synthetic and real-world data show that the proposed method improves over state-of-the-art continuous DAG-learning alternatives.

Andrei Buciulea, Madeline Navarro, Samuel Rey et al.

Graph learning is the fundamental task of estimating unknown graph connectivity from available data. Typical approaches assume that not only is all information available simultaneously but also that all nodes can be observed. However, in many real-world scenarios, data can neither be known completely nor obtained all at once. We present a novel method for online graph estimation that accounts for the presence of hidden nodes. We consider signals that are stationary on the underlying graph, which provides a model for the unknown connections to hidden nodes. We then formulate a convex optimization problem for graph learning from streaming, incomplete graph signals. We solve the proposed problem through an efficient proximal gradient algorithm that can run in real-time as data arrives sequentially. Additionally, we provide theoretical conditions under which our online algorithm is similar to batch-wise solutions. Through experimental results on synthetic and real-world data, we demonstrate the viability of our approach for online graph learning in the presence of missing observations.

Samuel Rey, Seyed Saman Saboksayr, Gonzalo Mateos

We address the problem of learning the topology of directed acyclic graphs (DAGs) from nodal observations, which adhere to a linear structural equation model. Recent advances framed the combinatorial DAG structure learning task as a continuous optimization problem, yet existing methods must contend with the complexities of non-convex optimization. To overcome this limitation, we assume that the latent DAG contains only non-negative edge weights. Leveraging this additional structure, we argue that cycles can be effectively characterized (and prevented) using a convex acyclicity function based on the log-determinant of the adjacency matrix. This convexity allows us to relax the task of learning the non-negative weighted DAG as an abstract convex optimization problem. We propose a DAG recovery algorithm based on the method of multipliers, that is guaranteed to return a global minimizer. Furthermore, we prove that in the infinite sample size regime, the convexity of our approach ensures the recovery of the true DAG structure. We empirically validate the performance of our algorithm in several reproducible synthetic-data test cases, showing that it outperforms state-of-the-art alternatives.

Samuel Rey, Luca Martino, Roberto San Millan et al.

Research on soundscapes has shifted the focus of environmental acoustics from noise levels to the perception of sounds, incorporating contextual factors. Soundscape emotion recognition (SER) models perception using a set of features, with arousal and valence commonly regarded as sufficient descriptors of affect. In this work, we blend \emph{graph learning} techniques with a novel \emph{information criterion} to develop a feature selection framework for SER. Specifically, we estimate a sparse graph representation of feature relations using linear structural equation models (SEM) tailored to the widely used Emo-Soundscapes dataset. The resulting graph captures the relations between input features and the two emotional outputs. To determine the appropriate level of sparsity, we propose a novel \emph{generalized elbow detector}, which provides both a point estimate and an uncertainty interval. We conduct an extensive evaluation of our methods, including visualizations of the inferred relations. While several of our findings align with previous studies, the graph representation also reveals a strong connection between arousal and valence, challenging common SER assumptions.

Samuel Rey, Gonzalo Mateos

This work aims to learn the directed acyclic graph (DAG) that captures the instantaneous dependencies underlying a multivariate time series. The observed data follow a linear structural vector autoregressive model (SVARM) with both instantaneous and time-lagged dependencies, where the instantaneous structure is modeled by a DAG to reflect potential causal relationships. While recent continuous relaxation approaches impose acyclicity through smooth constraint functions involving powers of the adjacency matrix, they lead to non-convex optimization problems that are challenging to solve. In contrast, we assume that the underlying DAG has only non-negative edge weights, and leverage this additional structure to impose acyclicity via a convex constraint. This enables us to cast the problem of non-negative DAG recovery from multivariate time-series data as a convex optimization problem in abstract form, which we solve using the method of multipliers. Crucially, the convex formulation guarantees global optimality of the solution. Finally, we assess the performance of the proposed method on synthetic time-series data, where it outperforms existing alternatives.

Samuel Rey, Hamed Ajorlou, Gonzalo Mateos

We develop a novel convolutional architecture tailored for learning from data defined over directed acyclic graphs (DAGs). DAGs can be used to model causal relationships among variables, but their nilpotent adjacency matrices pose unique challenges towards developing DAG signal processing and machine learning tools. To address this limitation, we harness recent advances offering alternative definitions of causal shifts and convolutions for signals on DAGs. We develop a novel convolutional graph neural network that integrates learnable DAG filters to account for the partial ordering induced by the graph topology, thus providing valuable inductive bias to learn effective representations of DAG-supported data. We discuss the salient advantages and potential limitations of the proposed DAG convolutional network (DCN) and evaluate its performance on two learning tasks using synthetic data: network diffusion estimation and source identification. DCN compares favorably relative to several baselines, showcasing its promising potential.

Victor M. Tenorio, Samuel Rey, Antonio G. Marques

Graph Neural Networks (GNNs) have emerged as a notorious alternative to address learning problems dealing with non-Euclidean datasets. However, although most works assume that the graph is perfectly known, the observed topology is prone to errors stemming from observational noise, graph-learning limitations, or adversarial attacks. If ignored, these perturbations may drastically hinder the performance of GNNs. To address this limitation, this work proposes a robust implementation of GNNs that explicitly accounts for the presence of perturbations in the observed topology. For any task involving GNNs, our core idea is to i) solve an optimization problem not only over the learnable parameters of the GNN but also over the true graph, and ii) augment the fitting cost with a term accounting for discrepancies on the graph. Specifically, we consider a convolutional GNN based on graph filters and follow an alternating optimization approach to handle the (non-differentiable and constrained) optimization problem by combining gradient descent and projected proximal updates. The resulting algorithm is not limited to a particular type of graph and is amenable to incorporating prior information about the perturbations. Finally, we assess the performance of the proposed method through several numerical experiments.

Victor M. Tenorio, Madeline Navarro, Samuel Rey et al.

Graph Neural Networks (GNNs) often struggle with heterophilic data, where connected nodes may have dissimilar labels, as they typically assume homophily and rely on local message passing. To address this, we propose creating alternative graph structures by linking nodes with similar structural attributes (e.g., role-based or global), thereby fostering higher label homophily on these new graphs. We theoretically prove that GNN performance can be improved by utilizing graphs with fewer false positive edges (connections between nodes of different classes) and that considering multiple graph views increases the likelihood of finding such beneficial structures. Building on these insights, we introduce Structure-Guided GNN (SG-GNN), an architecture that processes the original graph alongside the newly created structural graphs, adaptively learning to weigh their contributions. Extensive experiments on various benchmark datasets, particularly those with heterophilic characteristics, demonstrate that our SG-GNN achieves state-of-the-art or highly competitive performance, highlighting the efficacy of exploiting structural information to guide GNNs.

Victor M. Tenorio, Madeline Navarro, Samuel Rey et al.

Graph Neural Networks (GNNs) have emerged as a promising tool to handle data exhibiting an irregular structure. However, most GNN architectures perform well on homophilic datasets, where the labels of neighboring nodes are likely to be the same. In recent years, an increasing body of work has been devoted to the development of GNN architectures for heterophilic datasets, where labels do not exhibit this low-pass behavior. In this work, we create a new graph in which nodes are connected if they share structural characteristics, meaning a higher chance of sharing their labels, and then use this new graph in the GNN architecture. To do this, we compute the k-nearest neighbors graph according to distances between structural features, which are either (i) role-based, such as degree, or (ii) global, such as centrality measures. Experiments show that the labels are smoother in this newly defined graph and that the performance of GNN architectures improves when using this alternative structure.

Madeline Navarro, Andrei Buciulea, Samuel Rey et al.

We estimate fair graphs from graph-stationary nodal observations such that connections are not biased with respect to sensitive attributes. Edges in real-world graphs often exhibit preferences for connecting certain pairs of groups. Biased connections can not only exacerbate but even induce unfair treatment for downstream graph-based tasks. We therefore consider group and individual fairness for graphs corresponding to group- and node-level definitions, respectively. To evaluate the fairness of a given graph, we provide multiple bias metrics, including novel measurements in the spectral domain. Furthermore, we propose Fair Spectral Templates (FairSpecTemp), an optimization-based method with two variants for estimating fair graphs from stationary graph signals, a general model for graph data subsuming many existing ones. One variant of FairSpecTemp exploits commutativity properties of graph stationarity while directly constraining bias, while the other implicitly encourages fair estimates by restricting bias in the graph spectrum and is thus more flexible. Our methods enjoy high probability performance bounds, yielding a conditional tradeoff between fairness and accuracy. In particular, our analysis reveals that accuracy need not be sacrificed to recover fair graphs. We evaluate FairSpecTemp on synthetic and real-world data sets to illustrate its effectiveness and highlight the advantages of both variants of FairSpecTemp.

Andrea Cavallo, Samuel Rey, Antonio G. Marques et al.

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.

Sergio Rozada, Samuel Rey, Gonzalo Mateos et al.

Dynamic programming (DP) is a fundamental tool used across many engineering fields. The main goal of DP is to solve Bellman's optimality equations for a given Markov decision process (MDP). Standard methods like policy iteration exploit the fixed-point nature of these equations to solve them iteratively. However, these algorithms can be computationally expensive when the state-action space is large or when the problem involves long-term dependencies. Here we propose a new approach that unrolls and truncates policy iterations into a learnable parametric model dubbed BellNet, which we train to minimize the so-termed Bellman error from random value function initializations. Viewing the transition probability matrix of the MDP as the adjacency of a weighted directed graph, we draw insights from graph signal processing to interpret (and compactly re-parameterize) BellNet as a cascade of nonlinear graph filters. This fresh look facilitates a concise, transferable, and unifying representation of policy and value iteration, with an explicit handle on complexity during inference. Preliminary experiments conducted in a grid-like environment demonstrate that BellNet can effectively approximate optimal policies in a fraction of the iterations required by classical methods.

Samuel Rey, Hamed Ajorlou, Gonzalo Mateos

Directed acyclic graphs (DAGs) are central to science and engineering applications including causal inference, scheduling, and neural architecture search. In this work, we introduce the DAG Convolutional Network (DCN), a novel graph neural network (GNN) architecture designed specifically for convolutional learning from signals supported on DAGs. The DCN leverages causal graph filters to learn nodal representations that account for the partial ordering inherent to DAGs, a strong inductive bias does not present in conventional GNNs. Unlike prior art in machine learning over DAGs, DCN builds on formal convolutional operations that admit spectral-domain representations. We further propose the Parallel DCN (PDCN), a model that feeds input DAG signals to a parallel bank of causal graph-shift operators and processes these DAG-aware features using a shared multilayer perceptron. This way, PDCN decouples model complexity from graph size while maintaining satisfactory predictive performance. The architectures' permutation equivariance and expressive power properties are also established. Comprehensive numerical tests across several tasks, datasets, and experimental conditions demonstrate that (P)DCN compares favorably with state-of-the-art baselines in terms of accuracy, robustness, and computational efficiency. These results position (P)DCN as a viable framework for deep learning from DAG-structured data that is designed from first (graph) signal processing principles.

Samuel Rey, Ernesto Curbelo, Luca Martino et al.

This work addresses the problem of graph learning from data following a Gaussian Graphical Model (GGM) with a time-varying mean. Graphical Lasso (GL), the standard method for estimating sparse precision matrices, assumes that the observed data follows a zero-mean Gaussian distribution. However, this assumption is often violated in real-world scenarios where the mean evolves over time due to external influences, trends, or regime shifts. When the mean is not properly accounted for, applying GL directly can lead to estimating a biased precision matrix, hence hindering the graph learning task. To overcome this limitation, we propose Graphical Lasso with Adaptive Targeted Adaptive Importance Sampling (GL-ATAIS), an iterative method that jointly estimates the time-varying mean and the precision matrix. Our approach integrates Bayesian inference with frequentist estimation, leveraging importance sampling to obtain an estimate of the mean while using a regularized maximum likelihood estimator to infer the precision matrix. By iteratively refining both estimates, GL-ATAIS mitigates the bias introduced by time-varying means, leading to more accurate graph recovery. Our numerical evaluation demonstrates the impact of properly accounting for time-dependent means and highlights the advantages of GL-ATAIS over standard GL in recovering the true graph structure.

Madeline Navarro, Samuel Rey, Andrei Buciulea et al.

We propose estimating Gaussian graphical models (GGMs) that are fair with respect to sensitive nodal attributes. Many real-world models exhibit unfair discriminatory behavior due to biases in data. Such discrimination is known to be exacerbated when data is equipped with pairwise relationships encoded in a graph. Additionally, the effect of biased data on graphical models is largely underexplored. We thus introduce fairness for graphical models in the form of two bias metrics to promote balance in statistical similarities across nodal groups with different sensitive attributes. Leveraging these metrics, we present Fair GLASSO, a regularized graphical lasso approach to obtain sparse Gaussian precision matrices with unbiased statistical dependencies across groups. We also propose an efficient proximal gradient algorithm to obtain the estimates. Theoretically, we express the tradeoff between fair and accurate estimated precision matrices. Critically, this includes demonstrating when accuracy can be preserved in the presence of a fairness regularizer. On top of this, we study the complexity of Fair GLASSO and demonstrate that our algorithm enjoys a fast convergence rate. Our empirical validation includes synthetic and real-world simulations that illustrate the value and effectiveness of our proposed optimization problem and iterative algorithm.

Samuel Rey, Andrei Buciulea, Madeline Navarro et al.

Learning graphs from sets of nodal observations represents a prominent problem formally known as graph topology inference. However, current approaches are limited by typically focusing on inferring single networks, and they assume that observations from all nodes are available. First, many contemporary setups involve multiple related networks, and second, it is often the case that only a subset of nodes is observed while the rest remain hidden. Motivated by these facts, we introduce a joint graph topology inference method that models the influence of the hidden variables. Under the assumptions that the observed signals are stationary on the sought graphs and the graphs are closely related, the joint estimation of multiple networks allows us to exploit such relationships to improve the quality of the learned graphs. Moreover, we confront the challenging problem of modeling the influence of the hidden nodes to minimize their detrimental effect. To obtain an amenable approach, we take advantage of the particular structure of the setup at hand and leverage the similarity between the different graphs, which affects both the observed and the hidden nodes. To test the proposed method, numerical simulations over synthetic and real-world graphs are provided.

Victor M. Tenorio, Samuel Rey, Fernando Gama et al.

Graph convolutional neural networks (GCNNs) are popular deep learning architectures that, upon replacing regular convolutions with graph filters (GFs), generalize CNNs to irregular domains. However, classical GFs are prone to numerical errors since they consist of high-order polynomials. This problem is aggravated when several filters are applied in cascade, limiting the practical depth of GCNNs. To tackle this issue, we present the neighborhood graph filters (NGFs), a family of GFs that replaces the powers of the graph shift operator with $k$-hop neighborhood adjacency matrices. NGFs help to alleviate the numerical issues of traditional GFs, allow for the design of deeper GCNNs, and enhance the robustness to errors in the topology of the graph. To illustrate the advantage over traditional GFs in practical applications, we use NGFs in the design of deep neighborhood GCNNs to solve graph signal denoising and node classification problems over both synthetic and real-world data.

Samuel Rey, Santiago Segarra, Reinhard Heckel et al.

A fundamental problem in signal processing is to denoise a signal. While there are many well-performing methods for denoising signals defined on regular supports, such as images defined on two-dimensional grids of pixels, many important classes of signals are defined over irregular domains such as graphs. This paper introduces two untrained graph neural network architectures for graph signal denoising, provides theoretical guarantees for their denoising capabilities in a simple setup, and numerically validates the theoretical results in more general scenarios. The two architectures differ on how they incorporate the information encoded in the graph, with one relying on graph convolutions and the other employing graph upsampling operators based on hierarchical clustering. Each architecture implements a different prior over the targeted signals. To numerically illustrate the validity of the theoretical results and to compare the performance of the proposed architectures with other denoising alternatives, we present several experimental results with real and synthetic datasets.

Samuel Rey, Antonio G. Marques, Santiago Segarra

While deep convolutional architectures have achieved remarkable results in a gamut of supervised applications dealing with images and speech, recent works show that deep untrained non-convolutional architectures can also outperform state-of-the-art methods in several tasks such as image compression and denoising. Motivated by the fact that many contemporary datasets have an irregular structure different from a 1D/2D grid, this paper generalizes untrained and underparametrized non-convolutional architectures to signals defined over irregular domains represented by graphs. The proposed architecture consists of a succession of layers, each of them implementing an upsampling operator, a linear feature combination, and a scalar nonlinearity. A novel element is the incorporation of upsampling operators accounting for the structure of the supporting graph, which is achieved by considering a systematic graph coarsening approach based on hierarchical clustering. The numerical results carried out in synthetic and real-world datasets showcase that the reconstruction performance can improve drastically if the information of the supporting graph topology is taken into account.