arXiv subjects

Antonio G. Marques

Publications and source records attributed to Antonio G. Marques.

At least 55 records · Page 3Linked to original sources

arXiv2024

Tensor and Matrix Low-Rank Value-Function Approximation in Reinforcement Learning

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.

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arXiv2024

Matrix Low-Rank Trust Region Policy Optimization

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.

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arXiv2024

Matrix Low-Rank Approximation For Policy Gradient Methods

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.

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arXiv2024

Tensor Low-rank Approximation of Finite-horizon Value Functions

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.

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arXiv2024

A Multi-resolution Low-rank Tensor Decomposition

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.

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arXiv2024

Polynomial Graphical Lasso: Learning Edges from Gaussian Graph-Stationary Signals

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.

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arXiv2024

Mitigating Subpopulation Bias for Fair Network Topology Inference

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.

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arXiv2024

Multimodal Interpretable Data-Driven Models for Early Prediction of Antimicrobial Multidrug Resistance Using Multivariate Time-Series

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.

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arXiv2023

Learning graphs and simplicial complexes from data

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.

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arXiv2023

Robust Graph Neural Network based on Graph Denoising

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.

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arXiv2023

Blind Deconvolution of Sparse Graph Signals in the Presence of Perturbations

Blind deconvolution over graphs involves using (observed) output graph signals to obtain both the inputs (sources) as well as the filter that drives (models) the graph diffusion process. This is an ill-posed problem that requires additional assumptions, such as the sources being sparse, to be solvable. This paper addresses the blind deconvolution problem in the presence of imperfect graph information, where the observed graph is a perturbed version of the (unknown) true graph. While not having perfect knowledge of the graph is arguably more the norm than the exception, the body of literature on this topic is relatively small. This is partly due to the fact that translating the uncertainty about the graph topology to standard graph signal processing tools (e.g. eigenvectors or polynomials of the graph) is a challenging endeavor. To address this limitation, we propose an optimization-based estimator that solves the blind identification in the vertex domain, aims at estimating the inverse of the generating filter, and accounts explicitly for additive graph perturbations. Preliminary numerical experiments showcase the effectiveness and potential of the proposed algorithm.

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arXiv2023

Recovering Missing Node Features with Local Structure-based Embeddings

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.

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arXiv2023

Blind Demixing of Diffused Graph Signals

Using graphs to model irregular information domains is an effective approach to deal with some of the intricacies of contemporary (network) data. A key aspect is how the data, represented as graph signals, depend on the topology of the graph. Widely-used approaches assume that the observed signals can be viewed as outputs of graph filters (i.e., polynomials of a matrix representation of the graph) whose inputs have a particular structure. Diffused graph signals, which correspond to an originally sparse (node-localized) signal percolated through the graph via filtering, fall into this class. In that context, this paper deals with the problem of jointly identifying graph filters and separating their (sparse) input signals from a mixture of diffused graph signals, thus generalizing to the graph signal processing framework the classical blind demixing (blind source separation) of temporal and spatial signals. We first consider the scenario where the supporting graphs are different across the signals, providing a theorem for demixing feasibility along with probabilistic bounds on successful recovery. Additionally, an analysis of the degenerate problem of demixing with a single graph is also presented. Numerical experiments with synthetic and real-world graphs empirically illustrating the main theoretical findings close the paper.

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arXiv2023

Graph Signal Processing: History, Development, Impact, and Outlook

Graph signal processing (GSP) generalizes signal processing (SP) tasks to signals living on non-Euclidean domains whose structure can be captured by a weighted graph. Graphs are versatile, able to model irregular interactions, easy to interpret, and endowed with a corpus of mathematical results, rendering them natural candidates to serve as the basis for a theory of processing signals in more irregular domains. In this article, we provide an overview of the evolution of GSP, from its origins to the challenges ahead. The first half is devoted to reviewing the history of GSP and explaining how it gave rise to an encompassing framework that shares multiple similarities with SP. A key message is that GSP has been critical to develop novel and technically sound tools, theory, and algorithms that, by leveraging analogies with and the insights of digital SP, provide new ways to analyze, process, and learn from graph signals. In the second half, we shift focus to review the impact of GSP on other disciplines. First, we look at the use of GSP in data science problems, including graph learning and graph-based deep learning. Second, we discuss the impact of GSP on applications, including neuroscience and image and video processing. We conclude with a brief discussion of the emerging and future directions of GSP.

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arXiv2023

Graph Learning from Gaussian and Stationary Graph Signals

Graphs have become pervasive tools to represent information and datasets with irregular support. However, in many cases, the underlying graph is either unavailable or naively obtained, calling for more advanced methods to its estimation. Indeed, graph topology inference methods that estimate the network structure from a set of signal observations have a long and well established history. By assuming that the observations are both Gaussian and stationary in the sought graph, this paper proposes a new scheme to learn the network from nodal observations. Consideration of graph stationarity overcomes some of the limitations of the classical Graphical Lasso algorithm, which is constrained to a more specific class of graphical models. On the other hand, Gaussianity allows us to regularize the estimation, requiring less samples than in existing graph stationarity-based approaches. While the resultant estimation (optimization) problem is more complex and non-convex, we design an alternating convex approach able to find a stationary solution. Numerical tests with synthetic and real data are presented, and the performance of our approach is compared with existing alternatives.

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arXiv2023

Untrained Graph Neural Networks for Denoising

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.

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arXiv2022

Joint graph learning from Gaussian observations in the presence of hidden nodes

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.

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arXiv2022

Robust Graph Filter Identification and Graph Denoising from Signal Observations

When facing graph signal processing tasks, the workhorse assumption is that the graph describing the support of the signals is known. However, in many relevant applications the available graph suffers from observation errors and perturbations. As a result, any method relying on the graph topology may yield suboptimal results if those imperfections are ignored. Motivated by this, we propose a novel approach for handling perturbations on the links of the graph and apply it to the problem of robust graph filter (GF) identification from input-output observations. Different from existing works, we formulate a non-convex optimization problem that operates in the vertex domain and jointly performs GF identification and graph denoising. As a result, on top of learning the desired GF, an estimate of the graph is obtained as a byproduct. To handle the resulting bi-convex problem, we design an algorithm that blends techniques from alternating optimization and majorization minimization, showing its convergence to a stationary point. The second part of the paper i) generalizes the design to a robust setup where several GFs are jointly estimated, and ii) introduces an alternative algorithmic implementation that reduces the computational complexity. Finally, the detrimental influence of the perturbations and the benefits resulting from the robust approach are numerically analyzed over synthetic and real-world datasets, comparing them with other state-of-the-art alternatives.

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