Antonio G. Marques

Luis M. Lopez-Ramos, Antonio G. Marques, Javier Ramos

Successful deployment of cognitive radios requires efficient sensing of the spectrum and dynamic adaptation of the available resources according to the sensed (imperfect) information. While most works design these two tasks separately, in this paper we address them jointly. In particular, we investigate an overlay cognitive radio with multiple secondary users that access orthogonally a set of frequency bands originally devoted to primary users. The schemes are designed to minimize the cost of sensing, maximize the performance of the secondary users (weighted sum rate), and limit the probability of interfering the primary users. The joint design is addressed using dynamic programming and nonlinear optimization techniques. A two-step strategy that first finds the optimal resource allocation for any sensing scheme and then uses that solution as input to solve for the optimal sensing policy is implemented. The two-step strategy is optimal, gives rise to intuitive optimal policies, and entails a computational complexity much lower than that required to solve the original formulation.

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, 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.

Victor M. Tenorio, Madeline Navarro, Santiago Segarra et al.

Node features bolster graph-based learning when exploited jointly with network structure. However, a lack of nodal attributes is prevalent in graph data. We present a framework to recover completely missing node features for a set of graphs, where we only know the signals of a subset of graphs. Our approach incorporates prior information from both graph topology and existing nodal values. We demonstrate an example implementation of our framework where we assume that node features depend on local graph structure. Missing nodal values are estimated by aggregating known features from the most similar nodes. Similarity is measured through a node embedding space that preserves local topological features, which we train using a Graph AutoEncoder. We empirically show not only the accuracy of our feature estimation approach but also its value for downstream graph classification. Our success embarks on and implies the need to emphasize the relationship between node features and graph structure in graph-based learning.

Óscar Escudero-Arnanz, Cristina Soguero-Ruiz, Joaquín Álvarez-Rodríguez et al.

Antimicrobial Resistance represents a significant challenge in the Intensive Care Unit (ICU), where patients are at heightened risk of Multidrug-Resistant (MDR) infections-pathogens resistant to multiple antimicrobial agents. This study introduces a novel methodology that integrates Gated Recurrent Units (GRUs) with advanced intrinsic and post-hoc interpretability techniques for detecting the onset of MDR in patients across time. Within interpretability methods, we propose Explainable Artificial Intelligence (XAI) approaches to handle irregular Multivariate Time Series (MTS), introducing Irregular Time Shapley Additive Explanations (IT-SHAP), a modification of Shapley Additive Explanations designed for irregular MTS with Recurrent Neural Networks focused on temporal outputs. Our methodology aims to identify specific risk factors associated with MDR in ICU patients. GRU with Hadamard's attention demonstrated high initial specificity and increasing sensitivity over time, correlating with increased nosocomial infection risks during prolonged ICU stays. XAI analysis, enhanced by Hadamard attention and IT-SHAP, identified critical factors such as previous non-resistant cultures, specific antibiotic usage patterns, and hospital environment dynamics. These insights suggest that early detection of at-risk patients can inform interventions such as preventive isolation and customized treatments, significantly improving clinical outcomes. The proposed GRU model for temporal classification achieved an average Receiver Operating Characteristic Area Under the Curve of 78.27 +- 1.26 over time, indicating strong predictive performance. In summary, this study highlights the clinical utility of our methodology, which combines predictive accuracy with interpretability, thereby facilitating more effective healthcare interventions by professionals.

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.

Óscar Escudero-Arnanz, Antonio G. Marques, Inmaculada Mora-Jiménez et al.

Background and Objectives: Multidrug Resistance (MDR) is a critical global health issue, causing increased hospital stays, healthcare costs, and mortality. This study proposes an interpretable Machine Learning (ML) framework for MDR prediction, aiming for both accurate inference and enhanced explainability. Methods: Patients are modeled as Multivariate Time Series (MTS), capturing clinical progression and patient-to-patient interactions. Similarity among patients is quantified using MTS-based methods: descriptive statistics, Dynamic Time Warping, and Time Cluster Kernel. These similarity measures serve as inputs for MDR classification via Logistic Regression, Random Forest, and Support Vector Machines, with dimensionality reduction and kernel transformations improving model performance. For explainability, patient similarity networks are constructed from these metrics. Spectral clustering and t-SNE are applied to identify MDR-related subgroups and visualize high-risk clusters, enabling insight into clinically relevant patterns. Results: The framework was validated on ICU Electronic Health Records from the University Hospital of Fuenlabrada, achieving an AUC of 81%. It outperforms baseline ML and deep learning models by leveraging graph-based patient similarity. The approach identifies key risk factors -- prolonged antibiotic use, invasive procedures, co-infections, and extended ICU stays -- and reveals clinically meaningful clusters. Code and results are available at \https://github.com/oscarescuderoarnanz/DM4MTS. Conclusions: Patient similarity representations combined with graph-based analysis provide accurate MDR prediction and interpretable insights. This method supports early detection, risk factor identification, and patient stratification, highlighting the potential of explainable ML in critical care.

Sergio Rozada, Dongsheng Ding, Antonio G. Marques et al.

We study the problem of computing deterministic optimal policies for constrained Markov decision processes (MDPs) with continuous state and action spaces, which are widely encountered in constrained dynamical systems. Designing deterministic policy gradient methods in continuous state and action spaces is particularly challenging due to the lack of enumerable state-action pairs and the adoption of deterministic policies, hindering the application of existing policy gradient methods. To this end, we develop a deterministic policy gradient primal-dual method to find an optimal deterministic policy with non-asymptotic convergence. Specifically, we leverage regularization of the Lagrangian of the constrained MDP to propose a deterministic policy gradient primal-dual (D-PGPD) algorithm that updates the deterministic policy via a quadratic-regularized gradient ascent step and the dual variable via a quadratic-regularized gradient descent step. We prove that the primal-dual iterates of D-PGPD converge at a sub-linear rate to an optimal regularized primal-dual pair. We instantiate D-PGPD with function approximation and prove that the primal-dual iterates of D-PGPD converge at a sub-linear rate to an optimal regularized primal-dual pair, up to a function approximation error. Furthermore, we demonstrate the effectiveness of our method in two continuous control problems: robot navigation and fluid control. This appears to be the first work that proposes a deterministic policy search method for continuous-space constrained MDPs.

Jesus Fernandez-Bes, Jesus Cid-Sueiro, Antonio G. Marques

In this paper, we propose a novel censoring policy for energy-efficient transmissions in energy-harvesting sensors. The problem is formulated as an infinite-horizon Markov Decision Process (MDP). The objective to be optimized is the expected sum of the importance (utility) of all transmitted messages. Assuming that such importance can be evaluated at the transmitting node, we show that, under certain conditions on the battery model, the optimal censoring policy is a threshold function on the importance value. Specifically, messages are transmitted only if their importance is above a threshold whose value depends on the battery level. Exploiting this property, we propose a model-based stochastic scheme that approximates the optimal solution, with less computational complexity and faster convergence speed than a conventional Q-learning algorithm. Numerical experiments in single-hop and multi-hop networks confirm the analytical advantages of the proposed scheme.

Andrei Buciulea, Jiaxi Ying, Antonio G. Marques et al.

This paper introduces Polynomial Graphical Lasso (PGL), a new approach to learning graph structures from nodal signals. Our key contribution lies in modeling the signals as Gaussian and stationary on the graph, enabling the development of a graph-learning formulation that combines the strengths of graphical lasso with a more encompassing model. Specifically, we assume that the precision matrix can take any polynomial form of the sought graph, allowing for increased flexibility in modeling nodal relationships. Given the resulting complexity and nonconvexity of the resulting optimization problem, we (i) propose a low-complexity algorithm that alternates between estimating the graph and precision matrices, and (ii) characterize its convergence. We evaluate the performance of PGL through comprehensive numerical simulations using both synthetic and real data, demonstrating its superiority over several alternatives. Overall, this approach presents a significant advancement in graph learning and holds promise for various applications in graph-aware signal analysis and beyond.

Andrei Buciulea, Elvin Isufi, Geert Leus et al.

Graphs are widely used to represent complex information and signal domains with irregular support. Typically, the underlying graph topology is unknown and must be estimated from the available data. Common approaches assume pairwise node interactions and infer the graph topology based on this premise. In contrast, our novel method not only unveils the graph topology but also identifies three-node interactions, referred to in the literature as second-order simplicial complexes (SCs). We model signals using a graph autoregressive Volterra framework, enhancing it with structured graph Volterra kernels to learn SCs. We propose a mathematical formulation for graph and SC inference, solving it through convex optimization involving group norms and mask matrices. Experimental results on synthetic and real-world data showcase a superior performance for our approach compared to existing methods.

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.

Óscar Escudero-Arnanz, Cristina Soguero-Ruiz, Antonio G. Marques

In this paper, we present XST-GCNN (eXplainable Spatio-Temporal Graph Convolutional Neural Network), a novel architecture for processing heterogeneous and irregular Multivariate Time Series (MTS) data. Our approach captures temporal and feature dependencies within a unified spatio-temporal pipeline by leveraging a GCNN that uses a spatio-temporal graph aimed at optimizing predictive accuracy and interoperability. For graph estimation, we introduce techniques, including one based on the (heterogeneous) Gower distance. Once estimated, we propose two methods for graph construction: one based on the Cartesian product, treating temporal instants homogeneously, and another spatio-temporal approach with distinct graphs per time step. We also propose two GCNN architectures: a standard GCNN with a normalized adjacency matrix and a higher-order polynomial GCNN. In addition to accuracy, we emphasize explainability by designing an inherently interpretable model and performing a thorough interpretability analysis, identifying key feature-time combinations that drive predictions. We evaluate XST-GCNN using real-world Electronic Health Record data from University Hospital of Fuenlabrada to predict Multidrug Resistance (MDR) in ICU patients, a critical healthcare challenge linked to high mortality and complex treatments. Our architecture outperforms traditional models, achieving a mean ROC-AUC score of 81.03 +- 2.43. Furthermore, the interpretability analysis provides actionable insights into clinical factors driving MDR predictions, enhancing model transparency. This work sets a benchmark for tackling complex inference tasks with heterogeneous MTS, offering a versatile, interpretable solution for real-world applications.

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.

Reza Ramezanpour, Victor M. Tenorio, Antonio G. Marques et al.

Graphons, as limit objects of dense graph sequences, play a central role in the statistical analysis of network data. However, existing graphon estimation methods often struggle with scalability to large networks and resolution-independent approximation, due to their reliance on estimating latent variables or costly metrics such as the Gromov-Wasserstein distance. In this work, we propose a novel, scalable graphon estimator that directly recovers the graphon via moment matching, leveraging implicit neural representations (INRs). Our approach avoids latent variable modeling by training an INR--mapping coordinates to graphon values--to match empirical subgraph counts (i.e., moments) from observed graphs. This direct estimation mechanism yields a polynomial-time solution and crucially sidesteps the combinatorial complexity of Gromov-Wasserstein optimization. Building on foundational results, we establish a theoretical guarantee: when the observed subgraph motifs sufficiently represent those of the true graphon (a condition met with sufficiently large or numerous graph samples), the estimated graphon achieves a provable upper bound in cut distance from the ground truth. Additionally, we introduce MomentMixup, a data augmentation technique that performs mixup in the moment space to enhance graphon-based learning. Our graphon estimation method achieves strong empirical performance--demonstrating high accuracy on small graphs and superior computational efficiency on large graphs--outperforming state-of-the-art scalable estimators in 75\% of benchmark settings and matching them in the remaining cases. Furthermore, MomentMixup demonstrated improved graph classification accuracy on the majority of our benchmarks.

Sergio Rozada, Jose Luis Orejuela, Antonio G. Marques

We study the problem of learning optimal policies in finite-horizon Markov Decision Processes (MDPs) using low-rank reinforcement learning (RL) methods. In finite-horizon MDPs, the policies, and therefore the value functions (VFs) are not stationary. This aggravates the challenges of high-dimensional MDPs, as they suffer from the curse of dimensionality and high sample complexity. To address these issues, we propose modeling the VFs of finite-horizon MDPs as low-rank tensors, enabling a scalable representation that renders the problem of learning optimal policies tractable. We introduce an optimization-based framework for solving the Bellman equations with low-rank constraints, along with block-coordinate descent (BCD) and block-coordinate gradient descent (BCGD) algorithms, both with theoretical convergence guarantees. For scenarios where the system dynamics are unknown, we adapt the proposed BCGD method to estimate the VFs using sampled trajectories. Numerical experiments further demonstrate that the proposed framework reduces computational demands in controlled synthetic scenarios and more realistic resource allocation problems.

Sergio Rozada, Hoi-To Wai, Antonio G. Marques

Reinforcement learning (RL) aims to estimate the action to take given a (time-varying) state, with the goal of maximizing a cumulative reward function. Predominantly, there are two families of algorithms to solve RL problems: value-based and policy-based methods, with the latter designed to learn a probabilistic parametric policy from states to actions. Most contemporary approaches implement this policy using a neural network (NN). However, NNs usually face issues related to convergence, architectural suitability, hyper-parameter selection, and underutilization of the redundancies of the state-action representations (e.g. locally similar states). This paper postulates multi-linear mappings to efficiently estimate the parameters of the RL policy. More precisely, we leverage the PARAFAC decomposition to design tensor low-rank policies. The key idea involves collecting the policy parameters into a tensor and leveraging tensor-completion techniques to enforce low rank. We establish theoretical guarantees of the proposed methods for various policy classes and validate their efficacy through numerical experiments. Specifically, we demonstrate that tensor low-rank policy models reduce computational and sample complexities in comparison to NN models while achieving similar rewards.

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.

Victor M. Tenorio, Antonio G. Marques

Graph neural networks (GNNs) have emerged as a promising solution to deal with unstructured data, outperforming traditional deep learning architectures. However, most of the current GNN models are designed to work with a single graph, which limits their applicability in many real-world scenarios where multiple graphs may be involved. To address this limitation, we propose a novel graph-based deep learning architecture to handle tasks where two sets of signals exist, each defined on a different graph. First we consider the setting where the input is represented as a signal on top of one graph (input graph) and the output is a graph signal defined over a different graph (output graph). For this setup, we propose a three-block architecture where we first process the input data using a GNN that operates over the input graph, then apply a transformation function that operates in a latent space and maps the signals from the input to the output graph, and finally implement a second GNN that operates over the output graph. Our goal is not to propose a single specific definition for each of the three blocks, but rather to provide a flexible approach to solve tasks involving data defined on two graphs. The second part of the paper addresses a self-supervised setup, where the focus is not on the output space but on the underlying latent space and, inspired by Canonical Correlation Analysis, we seek informative representations of the data that can be leveraged to solve a downstream task. By leveraging information from multiple graphs, the proposed architecture can capture more intricate relationships between different entities in the data. We test this in several experimental setups using synthetic and real world datasets, and observe that the proposed architecture works better than traditional deep learning architectures, showcasing the importance of leveraging the information of the two graphs.

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.

Sergio Rozada, Vimal K. B., Andrea Cavallo et al.

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.

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, 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.

Sergio Rozada, Santiago Paternain, Juan Andres Bazerque et al.

In pursuit of reinforcement learning systems that could train in physical environments, we investigate multi-task approaches as a means to alleviate the need for massive data acquisition. In a tabular scenario where the Q-functions are collected across tasks, we model our learning problem as optimizing a higher order tensor structure. Recognizing that close-related tasks may require similar actions, our proposed method imposes a low-rank condition on this aggregated Q-tensor. The rationale behind this approach to multi-task learning is that the low-rank structure enforces the notion of similarity, without the need to explicitly prescribe which tasks are similar, but inferring this information from a reduced amount of data simultaneously with the stochastic optimization of the Q-tensor. The efficiency of our low-rank tensor approach to multi-task learning is demonstrated in two numerical experiments, first in a benchmark environment formed by a collection of inverted pendulums, and then into a practical scenario involving multiple wireless communication devices.

Liuyuan Jiang, Quan Xiao, Victor M. Tenorio et al.

Interest in bilevel optimization has grown in recent years, partially due to its applications to tackle challenging machine-learning problems. Several exciting recent works have been centered around developing efficient gradient-based algorithms that can solve bilevel optimization problems with provable guarantees. However, the existing literature mainly focuses on bilevel problems either without constraints, or featuring only simple constraints that do not couple variables across the upper and lower levels, excluding a range of complex applications. Our paper studies this challenging but less explored scenario and develops a (fully) first-order algorithm, which we term BLOCC, to tackle BiLevel Optimization problems with Coupled Constraints. We establish rigorous convergence theory for the proposed algorithm and demonstrate its effectiveness on two well-known real-world applications - hyperparameter selection in support vector machine (SVM) and infrastructure planning in transportation networks using the real data from the city of Seville.

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.

Sergio Martínez-Agüero, Antonio G. Marques, Inmaculada Mora-Jiménez et al.

Electronic health records (EHR) is an inherently multimodal register of the patient's health status characterized by static data and multivariate time series (MTS). While MTS are a valuable tool for clinical prediction, their fusion with other data modalities can possibly result in more thorough insights and more accurate results. Deep neural networks (DNNs) have emerged as fundamental tools for identifying and defining underlying patterns in the healthcare domain. However, fundamental improvements in interpretability are needed for DNN models to be widely used in the clinical setting. In this study, we present an approach built on a collection of interpretable multimodal data-driven models that may anticipate and understand the emergence of antimicrobial multidrug resistance (AMR) germs in the intensive care unit (ICU) of the University Hospital of Fuenlabrada (Madrid, Spain). The profile and initial health status of the patient are modeled using static variables, while the evolution of the patient's health status during the ICU stay is modeled using several MTS, including mechanical ventilation and antibiotics intake. The multimodal DNNs models proposed in this paper include interpretable principles in addition to being effective at predicting AMR and providing an explainable prediction support system for AMR in the ICU. Furthermore, our proposed methodology based on multimodal models and interpretability schemes can be leveraged in additional clinical problems dealing with EHR data, broadening the impact and applicability of our results.

Sergio Rozada, Santiago Paternain, Antonio G. Marques

Value-function (VF) approximation is a central problem in Reinforcement Learning (RL). Classical non-parametric VF estimation suffers from the curse of dimensionality. As a result, parsimonious parametric models have been adopted to approximate VFs in high-dimensional spaces, with most efforts being focused on linear and neural-network-based approaches. Differently, this paper puts forth a a parsimonious non-parametric approach, where we use stochastic low-rank algorithms to estimate the VF matrix in an online and model-free fashion. Furthermore, as VFs tend to be multi-dimensional, we propose replacing the classical VF matrix representation with a tensor (multi-way array) representation and, then, use the PARAFAC decomposition to design an online model-free tensor low-rank algorithm. Different versions of the algorithms are proposed, their complexity is analyzed, and their performance is assessed numerically using standardized RL environments.

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.

Sergio Rozada, Victor Tenorio, Antonio G. Marques

Value functions are central to Dynamic Programming and Reinforcement Learning but their exact estimation suffers from the curse of dimensionality, challenging the development of practical value-function (VF) estimation algorithms. Several approaches have been proposed to overcome this issue, from non-parametric schemes that aggregate states or actions to parametric approximations of state and action VFs via, e.g., linear estimators or deep neural networks. Relevantly, several high-dimensional state problems can be well-approximated by an intrinsic low-rank structure. Motivated by this and leveraging results from low-rank optimization, this paper proposes different stochastic algorithms to estimate a low-rank factorization of the $Q(s, a)$ matrix. This is a non-parametric alternative to VF approximation that dramatically reduces the computational and sample complexities relative to classical $Q$-learning methods that estimate $Q(s,a)$ separately for each state-action pair.

Madeline Navarro, Yuhao Wang, Antonio G. Marques et al.

Inferring graph structure from observations on the nodes is an important and popular network science task. Departing from the more common inference of a single graph and motivated by social and biological networks, we study the problem of jointly inferring multiple graphs from the observation of signals at their nodes (graph signals), which are assumed to be stationary in the sought graphs. From a mathematical point of view, graph stationarity implies that the mapping between the covariance of the signals and the sparse matrix representing the underlying graph is given by a matrix polynomial. A prominent example is that of Markov random fields, where the inverse of the covariance yields the sparse matrix of interest. From a modeling perspective, stationary graph signals can be used to model linear network processes evolving on a set of (not necessarily known) networks. Leveraging that matrix polynomials commute, a convex optimization method along with sufficient conditions that guarantee the recovery of the true graphs are provided when perfect covariance information is available. Particularly important from an empirical viewpoint, we provide high-probability bounds on the recovery error as a function of the number of signals observed and other key problem parameters. Numerical experiments using synthetic and real-world data demonstrate the effectiveness of the proposed method with perfect covariance information as well as its robustness in the noisy regime.

Vassilis N. Ioannidis, Antonio G. Marques, Georgios B. Giannakis

The era of "data deluge" has sparked renewed interest in graph-based learning methods and their widespread applications ranging from sociology and biology to transportation and communications. In this context of graph-aware methods, the present paper introduces a tensor-graph convolutional network (TGCN) for scalable semi-supervised learning (SSL) from data associated with a collection of graphs, that are represented by a tensor. Key aspects of the novel TGCN architecture are the dynamic adaptation to different relations in the tensor graph via learnable weights, and the consideration of graph-based regularizers to promote smoothness and alleviate over-parameterization. The ultimate goal is to design a powerful learning architecture able to: discover complex and highly nonlinear data associations, combine (and select) multiple types of relations, scale gracefully with the graph size, and remain robust to perturbations on the graph edges. The proposed architecture is relevant not only in applications where the nodes are naturally involved in different relations (e.g., a multi-relational graph capturing family, friendship and work relations in a social network), but also in robust learning setups where the graph entails a certain level of uncertainty, and the different tensor slabs correspond to different versions (realizations) of the nominal graph. Numerical tests showcase that the proposed architecture achieves markedly improved performance relative to standard GCNs, copes with state-of-the-art adversarial attacks, and leads to remarkable SSL performance over protein-to-protein interaction networks.

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.

Luana Ruiz, Fernando Gama, Antonio G. Marques et al.

Graph signals are signals with an irregular structure that can be described by a graph. Graph neural networks (GNNs) are information processing architectures tailored to these graph signals and made of stacked layers that compose graph convolutional filters with nonlinear activation functions. Graph convolutions endow GNNs with invariance to permutations of the graph nodes' labels. In this paper, we consider the design of trainable nonlinear activation functions that take into consideration the structure of the graph. This is accomplished by using graph median filters and graph max filters, which mimic linear graph convolutions and are shown to retain the permutation invariance of GNNs. We also discuss modifications to the backpropagation algorithm necessary to train local activation functions. The advantages of localized activation function architectures are demonstrated in four numerical experiments: source localization on synthetic graphs, authorship attribution of 19th century novels, movie recommender systems and scientific article classification. In all cases, localized activation functions are shown to improve model capacity.

Alireza Sadeghi, Fatemeh Sheikholeslami, Antonio G. Marques et al.

Small base stations (SBs) of fifth-generation (5G) cellular networks are envisioned to have storage devices to locally serve requests for reusable and popular contents by \emph{caching} them at the edge of the network, close to the end users. The ultimate goal is to shift part of the predictable load on the back-haul links, from on-peak to off-peak periods, contributing to a better overall network performance and service experience. To enable the SBs with efficient \textit{fetch-cache} decision-making schemes operating in dynamic settings, this paper introduces simple but flexible generic time-varying fetching and caching costs, which are then used to formulate a constrained minimization of the aggregate cost across files and time. Since caching decisions per time slot influence the content availability in future slots, the novel formulation for optimal fetch-cache decisions falls into the class of dynamic programming. Under this generic formulation, first by considering stationary distributions for the costs and file popularities, an efficient reinforcement learning-based solver known as value iteration algorithm can be used to solve the emerging optimization problem. Later, it is shown that practical limitations on cache capacity can be handled using a particular instance of the generic dynamic pricing formulation. Under this setting, to provide a light-weight online solver for the corresponding optimization, the well-known reinforcement learning algorithm, $Q$-learning, is employed to find optimal fetch-cache decisions. Numerical tests corroborating the merits of the proposed approach wrap up the paper.

Vassilis N. Ioannidis, Antonio G. Marques, Georgios B. Giannakis

The era of data deluge has sparked the interest in graph-based learning methods in a number of disciplines such as sociology, biology, neuroscience, or engineering. In this paper, we introduce a graph recurrent neural network (GRNN) for scalable semi-supervised learning from multi-relational data. Key aspects of the novel GRNN architecture are the use of multi-relational graphs, the dynamic adaptation to the different relations via learnable weights, and the consideration of graph-based regularizers to promote smoothness and alleviate over-parametrization. Our ultimate goal is to design a powerful learning architecture able to: discover complex and highly non-linear data associations, combine (and select) multiple types of relations, and scale gracefully with respect to the size of the graph. Numerical tests with real data sets corroborate the design goals and illustrate the performance gains relative to competing alternatives.

Luana Ruiz, Fernando Gama, Antonio G. Marques et al.

Graph neural networks (GNNs) have been shown to replicate convolutional neural networks' (CNNs) superior performance in many problems involving graphs. By replacing regular convolutions with linear shift-invariant graph filters (LSI-GFs), GNNs take into account the (irregular) structure of the graph and provide meaningful representations of network data. However, LSI-GFs fail to encode local nonlinear graph signal behavior, and so do regular activation functions, which are nonlinear but pointwise. To address this issue, we propose median activation functions with support on graph neighborhoods instead of individual nodes. A GNN architecture with a trainable multirresolution version of this activation function is then tested on synthetic and real-word datasets, where we show that median activation functions can improve GNN capacity with marginal increase in complexity.

Fernando Gama, Antonio G. Marques, Geert Leus et al.

Two architectures that generalize convolutional neural networks (CNNs) for the processing of signals supported on graphs are introduced. We start with the selection graph neural network (GNN), which replaces linear time invariant filters with linear shift invariant graph filters to generate convolutional features and reinterprets pooling as a possibly nonlinear subsampling stage where nearby nodes pool their information in a set of preselected sample nodes. A key component of the architecture is to remember the position of sampled nodes to permit computation of convolutional features at deeper layers. The second architecture, dubbed aggregation GNN, diffuses the signal through the graph and stores the sequence of diffused components observed by a designated node. This procedure effectively aggregates all components into a stream of information having temporal structure to which the convolution and pooling stages of regular CNNs can be applied. A multinode version of aggregation GNNs is further introduced for operation in large scale graphs. An important property of selection and aggregation GNNs is that they reduce to conventional CNNs when particularized to time signals reinterpreted as graph signals in a circulant graph. Comparative numerical analyses are performed in a source localization application over synthetic and real-world networks. Performance is also evaluated for an authorship attribution problem and text category classification. Multinode aggregation GNNs are consistently the best performing GNN architecture.

Fernando Gama, Antonio G. Marques, Alejandro Ribeiro et al.

Superior performance and ease of implementation have fostered the adoption of Convolutional Neural Networks (CNNs) for a wide array of inference and reconstruction tasks. CNNs implement three basic blocks: convolution, pooling and pointwise nonlinearity. Since the two first operations are well-defined only on regular-structured data such as audio or images, application of CNNs to contemporary datasets where the information is defined in irregular domains is challenging. This paper investigates CNNs architectures to operate on signals whose support can be modeled using a graph. Architectures that replace the regular convolution with a so-called linear shift-invariant graph filter have been recently proposed. This paper goes one step further and, under the framework of multiple-input multiple-output (MIMO) graph filters, imposes additional structure on the adopted graph filters, to obtain three new (more parsimonious) architectures. The proposed architectures result in a lower number of model parameters, reducing the computational complexity, facilitating the training, and mitigating the risk of overfitting. Simulations show that the proposed simpler architectures achieve similar performance as more complex models.

Fernando Gama, Geert Leus, Antonio G. Marques et al.

Convolutional neural networks (CNNs) are being applied to an increasing number of problems and fields due to their superior performance in classification and regression tasks. Since two of the key operations that CNNs implement are convolution and pooling, this type of networks is implicitly designed to act on data described by regular structures such as images. Motivated by the recent interest in processing signals defined in irregular domains, we advocate a CNN architecture that operates on signals supported on graphs. The proposed design replaces the classical convolution not with a node-invariant graph filter (GF), which is the natural generalization of convolution to graph domains, but with a node-varying GF. This filter extracts different local features without increasing the output dimension of each layer and, as a result, bypasses the need for a pooling stage while involving only local operations. A second contribution is to replace the node-varying GF with a hybrid node-varying GF, which is a new type of GF introduced in this paper. While the alternative architecture can still be run locally without requiring a pooling stage, the number of trainable parameters is smaller and can be rendered independent of the data dimension. Tests are run on a synthetic source localization problem and on the 20NEWS dataset.

Antonio G. Marques, Santiago Segarra, Geert Leus et al.

Stationarity is a cornerstone property that facilitates the analysis and processing of random signals in the time domain. Although time-varying signals are abundant in nature, in many practical scenarios the information of interest resides in more irregular graph domains. This lack of regularity hampers the generalization of the classical notion of stationarity to graph signals. The contribution in this paper is twofold. Firstly, we propose a definition of weak stationarity for random graph signals that takes into account the structure of the graph where the random process takes place, while inheriting many of the meaningful properties of the classical definition in the time domain. Our definition requires that stationary graph processes can be modeled as the output of a linear graph filter applied to a white input. We will show that this is equivalent to requiring the correlation matrix to be diagonalized by the graph Fourier transform. Secondly, we analyze the properties of the power spectral density and propose a number of methods to estimate it. We start with nonparametric approaches, including periodograms, window-based average periodograms, and filter banks. We then shift the focus to parametric approaches, discussing the estimation of moving-average (MA), autoregressive (AR) and ARMA processes. Finally, we illustrate the power spectral density estimation in synthetic and real-world graphs.

Santiago Segarra, Antonio G. Marques, Alejandro Ribeiro

We study the design of graph filters to implement arbitrary linear transformations between graph signals. Graph filters can be represented by matrix polynomials of the graph-shift operator, which captures the structure of the graph and is assumed to be given. Thus, graph-filter design consists in choosing the coefficients of these polynomials (known as filter coefficients) to resemble desired linear transformations. Due to the local structure of the graph-shift operator, graph filters can be implemented distributedly across nodes, making them suitable for networked settings. We determine spectral conditions under which a specific linear transformation can be implemented perfectly using graph filters. Furthermore, for the cases where perfect implementation is infeasible, the design of optimal approximations for different error metrics is analyzed. We introduce the notion of a node-variant graph filter, which allows the simultaneous implementation of multiple (regular) graph filters in different nodes of the graph. This additional flexibility enables the design of more general operators without undermining the locality in implementation. Perfect and approximate implementation of network operators is also studied for node-variant graph filters. We demonstrate the practical relevance of the developed framework by studying in detail the application of graph filters to the problems of finite-time consensus and analog network coding. Finally, we present additional numerical experiments comparing the performance of node-invariant and node-variant filters when approximating arbitrary linear network operators.

Luis M. Lopez-Ramos, Vassilis Kekatos, Antonio G. Marques et al.

Smart distribution grids should efficiently integrate stochastic renewable resources while effecting voltage regulation. The design of energy management schemes is challenging, one of the reasons being that energy management is a multistage problem where decisions are not all made at the same timescale and must account for the variability during real-time operation. The joint dispatch of slow- and fast-timescale controls in a smart distribution grid is considered here. The substation voltage, the energy exchanged with a main grid, and the generation schedules for small diesel generators have to be decided on a slow timescale; whereas optimal photovoltaic inverter setpoints are found on a more frequent basis. While inverter and looser voltage regulation limits are imposed at all times, tighter bus voltage constraints are enforced on the average or in probability, thus enabling more efficient renewable integration. Upon reformulating the two-stage grid dispatch as a stochastic convex-concave problem, two distribution-free schemes are put forth. An average dispatch algorithm converges provably to the optimal two-stage decisions via a sequence of convex quadratic programs. Its non-convex probabilistic alternative entails solving two slightly different convex problems and is numerically shown to converge. Numerical tests on a real-world distribution feeder verify that both novel data-driven schemes yield lower costs over competing alternatives.