Search

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.

This work presents a low-rank tensor model for multi-dimensional Markov chains. A common approach to simplify the dynamical behavior of a Markov chain is to impose low-rankness on the transition probability matrix. Inspired by the success of these matrix techniques, we present low-rank tensors for representing transition probabilities on multi-dimensional state spaces. Through tensor decomposition, we provide a connection between our method and classical probabilistic models. Moreover, our proposed model yields a parsimonious representation with fewer parameters than matrix-based approaches. Unlike these methods, which impose low-rankness uniformly across all states, our tensor method accounts for the multi-dimensionality of the state space. We also propose an optimization-based approach to estimate a Markov model as a low-rank tensor. Our optimization problem can be solved by the alternating direction method of multipliers (ADMM), which enjoys convergence to a stationary solution. We empirically demonstrate that our tensor model estimates Markov chains more efficiently than conventional techniques, requiring both fewer samples and parameters. We perform numerical simulations for both a synthetic low-rank Markov chain and a real-world example with New York City taxi data, showcasing the advantages of multi-dimensionality for modeling state spaces.

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.

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.

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.

This paper introduces a probabilistic approach for tracking the dynamics of unweighted and directed graphs using state-space models (SSMs). Unlike conventional topology inference methods that assume static graphs and generate point-wise estimates, our method accounts for dynamic changes in the network structure over time. We model the network at each timestep as the state of the SSM, and use observations to update beliefs that quantify the probability of the network being in a particular state. Then, by considering the dynamics of transition and observation models through the update and prediction steps, respectively, the proposed method can incorporate the information of real-time graph signals into the beliefs. These beliefs provide a probability distribution of the network at each timestep, being able to provide both an estimate for the network and the uncertainty it entails. Our approach is evaluated through experiments with synthetic and real-world networks. The results demonstrate that our method effectively estimates network states and accounts for the uncertainty in the data, outperforming traditional techniques such as recursive least squares., Comment: Submitted to the 2025 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2025)

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 for constrained MDPs. 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. To the best of our knowledge, this appears to be the first work that proposes a deterministic policy search method for continuous-space constrained MDPs.

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.

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., Comment: In this version, we have made the following updates: (1) Added a sensitivity analysis of the algorithm's hyperparameters (stepsize and penalty constant) in Appendix G. (2) Included a computational complexity analysis and comparison in Appendix H. (3) Explicitly stated the inner-loop stepsizes in Remarks 2 and 3

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.

The (efficient and parsimonious) decomposition of higher-order tensors is a fundamental problem with numerous applications in a variety of fields. Several methods have been proposed in the literature to that end, with the Tucker and PARAFAC decompositions being the most prominent ones. Inspired by the latter, in this work we propose a multi-resolution low-rank tensor decomposition to describe (approximate) a tensor in a hierarchical fashion. The central idea of the decomposition is to recast the tensor into \emph{multiple} lower-dimensional tensors to exploit the structure at different levels of resolution. The method is first explained, an alternating least squares algorithm is discussed, and preliminary simulations illustrating the potential practical relevance are provided.

The goal of reinforcement learning is estimating a policy that maps states to actions and maximizes the cumulative reward of a Markov Decision Process (MDP). This is oftentimes achieved by estimating first the optimal (reward) value function (VF) associated with each state-action pair. When the MDP has an infinite horizon, the optimal VFs and policies are stationary under mild conditions. However, in finite-horizon MDPs, the VFs (hence, the policies) vary with time. This poses a challenge since the number of VFs to estimate grows not only with the size of the state-action space but also with the time horizon. This paper proposes a non-parametric low-rank stochastic algorithm to approximate the VFs of finite-horizon MDPs. First, we represent the (unknown) VFs as a multi-dimensional array, or tensor, where time is one of the dimensions. Then, we use rewards sampled from the MDP to estimate the optimal VFs. More precisely, we use the (truncated) PARAFAC decomposition to design an online low-rank algorithm that recovers the entries of the tensor of VFs. The size of the low-rank PARAFAC model grows additively with respect to each of its dimensions, rendering our approach efficient, as demonstrated via numerical experiments.

Estimating a policy that maps states to actions is a central problem in reinforcement learning. Traditionally, policies are inferred from the so called value functions (VFs), but exact VF computation suffers from the curse of dimensionality. Policy gradient (PG) methods bypass this by learning directly a parametric stochastic policy. Typically, the parameters of the policy are estimated using neural networks (NNs) tuned via stochastic gradient descent. However, finding adequate NN architectures can be challenging, and convergence issues are common as well. In this paper, we put forth low-rank matrix-based models to estimate efficiently the parameters of PG algorithms. We collect the parameters of the stochastic policy into a matrix, and then, we leverage matrix-completion techniques to promote (enforce) low rank. We demonstrate via numerical studies how low-rank matrix-based policy models reduce the computational and sample complexities relative to NN models, while achieving a similar aggregated reward.

Most methods in reinforcement learning use a Policy Gradient (PG) approach to learn a parametric stochastic policy that maps states to actions. The standard approach is to implement such a mapping via a neural network (NN) whose parameters are optimized using stochastic gradient descent. However, PG methods are prone to large policy updates that can render learning inefficient. Trust region algorithms, like Trust Region Policy Optimization (TRPO), constrain the policy update step, ensuring monotonic improvements. This paper introduces low-rank matrix-based models as an efficient alternative for estimating the parameters of TRPO algorithms. By gathering the stochastic policy's parameters into a matrix and applying matrix-completion techniques, we promote and enforce low rank. Our numerical studies demonstrate that low-rank matrix-based policy models effectively reduce both computational and sample complexities compared to NN models, while maintaining comparable aggregated rewards.

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.

We consider fair network topology inference from nodal observations. Real-world networks often exhibit biased connections based on sensitive nodal attributes. Hence, different subpopulations of nodes may not share or receive information equitably. We thus propose an optimization-based approach to accurately infer networks while discouraging biased edges. To this end, we present bias metrics that measure topological demographic parity to be applied as convex penalties, suitable for most optimization-based graph learning methods. Moreover, we encourage equitable treatment for any number of subpopulations of differing sizes. We validate our method on synthetic and real-world simulations using networks with both biased and unbiased connections.

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.

We propose a novel algorithm for the support estimation of partially known Gaussian graphical models that incorporates prior information about the underlying graph. In contrast to classical approaches that provide a point estimate based on a maximum likelihood or a maximum a posteriori criterion using (simple) priors on the precision matrix, we consider a prior on the graph and rely on annealed Langevin diffusion to generate samples from the posterior distribution. Since the Langevin sampler requires access to the score function of the underlying graph prior, we use graph neural networks to effectively estimate the score from a graph dataset (either available beforehand or generated from a known distribution). Numerical experiments demonstrate the benefits of our approach., Comment: 17 pages, 7 figures, AISTATS 2024