Learning the graph Laplacian from observed data is one of the most investigated and fundamental tasks in Graph Signal Processing (GSP). Different variants of the Laplacian, such as the combinatorial, signless or signed Laplacians have been considered depending on the type of features to be extracted from the data. The main contribution of this paper is the introduction of a parametric Laplacian, called the deformed Laplacian, defined as a quadratic matrix polynomial that provides a parametric dictionary for graph signal processing. The deformed Laplacian can be interpreted as the generator of a parametric linear reaction-diffusion dynamics on graphs, capturing the interplay between diffusive coupling and nodal reaction effects. It is a parametric polynomial matrix that enables the design of novel topological operators tailored to both the underlying graph structure and the observed signals. Interestingly, we show that several Laplacian variants proposed in the literature arise as special cases of the deformed Laplacian. We then develop a method to jointly learn the deformed Laplacian and the graph signals from data, showing how its use improves signal representation across a broad
Our goal in this paper is to leverage the potential of the topological signal processing (TSP) framework for analyzing brain networks. Representing brain data as signals over simplicial complexes allows us to capture higher-order relationships within brain regions of interest (ROIs). Here, we focus on learning the underlying brain topology from observed neural signals using two distinct inference strategies. The first method relies on higher-order statistical metrics to infer multiway relationships among ROIs. The second method jointly learns the brain topology and sparse signal representations, of both the solenoidal and harmonic components of the signals, by minimizing the total variation along triangles and the data-fitting errors. Leveraging the properties of solenoidal and irrotational signals, and their physical interpretations, we extract functional connectivity features from brain topologies and uncover new insights into functional organization patterns. This allows us to associate brain functional connectivity (FC) patterns of conservative signals with well-known functional segregation and integration properties. Our findings align with recent neuroscience research, sugges
Signal processing stands as a pillar of classical computation and modern information technology, applicable to both analog and digital signals. Recently, advancements in quantum information science have suggested that quantum signal processing (QSP) can enable more powerful signal processing capabilities. However, the developments in QSP have primarily leveraged \emph{digital} quantum resources, such as discrete-variable (DV) systems like qubits, rather than \emph{analog} quantum resources, such as continuous-variable (CV) systems like quantum oscillators. Consequently, there remains a gap in understanding how signal processing can be performed on hybrid CV-DV quantum computers. Here we address this gap by developing a new paradigm of mixed analog-digital QSP. We demonstrate the utility of this paradigm by showcasing how it naturally enables analog-digital conversion of quantum signals -- specifically, the transfer of states between DV and CV quantum systems. We then show that such quantum analog-digital conversion enables new implementations of quantum algorithms on CV-DV hardware. This is exemplified by realizing the quantum Fourier transform of a state encoded on qubits via the
We present an uncertainty principle for graph signals in the vertex-time domain, unifying the classical time-frequency and graph uncertainty principles within a single framework. By defining vertex-time and spectral-frequency spreads, we quantify signal localization across these domains. Our framework identifies a class of signals that achieve maximum concentration in both the spatial and temporal domains. These signals serve as fundamental atoms for a new vertex-time dictionary, enhancing signal reconstruction under practical constraints, such as intermittent data commonly encountered in sensor and social networks. Furthermore, we introduce a novel graph topology inference method leveraging the uncertainty principle. Numerical experiments on synthetic and real datasets validate the effectiveness of our approach, demonstrating improved reconstruction accuracy, greater robustness to noise, and enhanced graph learning performance compared to existing methods.
Our goal in this paper is to apply the topological signal processing (TSP) framework to the analysis of 3D Point Clouds (PCs) represented on simplicial complexes. Building on Discrete Exterior Calculus (DEC) theory for vector fields, we introduce higher-order Laplacian operators that enable the processing of signals over triangular meshes. Unlike traditional approaches, the proposed approach allows us to characterize both color attributes, modeled as 3D vectors on nodes, and geometry, modeled as 3D vectors on the barycenter of each triangle. Then, we show as TSP tools may efficiently be used to sample, recover and filter PCs attributes treating them as edge signals. Numerical results on synthetic PCs demonstrate accurate color reconstruction with robustness to sparse data and geometry refinement in the case of noisy PC coordinates. The proposed approach provides a topology-based representation to characterize the geometry and attributes of PCs.
The proliferation of capable and efficient machine learning (ML) models marks one of the strongest methodological shifts in signal processing (SP) in its nearly 100-year history. ML models support the development of SP systems that represent complex, nonlinear relationships with high predictive accuracy. Adapting these models often requires sequential inference, which differs both theoretically and methodologically from the usual paradigm of ML, where data are often assumed independent and identically distributed. Gaussian processes (GPs) are a flexible yet principled framework for modeling random functions, and they have become increasingly relevant to SP as statistical and ML methods assume a more prominent role. We provide a self-contained, tutorial-style overview of GPs, with a particular focus on recent methodological advances in sequential, incremental, or streaming inference. We introduce these techniques from a signal-processing perspective while bridging them to recent advances in ML. Many of the developments we survey have direct applications to state-space modeling, sequential regression and forecasting, anomaly detection in time series, sequential Bayesian optimization,
This paper presents an algebraic theory of linear signal processing. At the core of algebraic signal processing is the concept of a linear signal model defined as a triple (A, M, phi), where familiar concepts like the filter space and the signal space are cast as an algebra A and a module M, respectively, and phi generalizes the concept of the z-transform to bijective linear mappings from a vector space of, e.g., signal samples, into the module M. A signal model provides the structure for a particular linear signal processing application, such as infinite and finite discrete time, or infinite or finite discrete space, or the various forms of multidimensional linear signal processing. As soon as a signal model is chosen, basic ingredients follow, including the associated notions of filtering, spectrum, and Fourier transform. The shift operator is a key concept in the algebraic theory: it is the generator of the algebra of filters A. Once the shift is chosen, a well-defined methodology leads to the associated signal model. Different shifts correspond to infinite and finite time models with associated infinite and finite z-transforms, and to infinite and finite space models with assoc
One of the key challenges in many research fields is uncovering how different interconnected systems interact within complex networks, typically represented as multi-layer networks. Capturing the intra- and cross-layer interactions among different domains for analysis and processing calls for topological algebraic descriptors capable of localizing the homologies of different domains, at different scales, according to the learning task. Our first contribution in this paper is to introduce the Cell MultiComplexes (CMCs), which are novel topological spaces that enable the representation of higher-order interactions among interconnected cell complexes. We introduce cross-Laplacian operators as powerful algebraic descriptors of CMC spaces able to capture different topological invariants, whether global or local, at different resolutions. Using the eigenvectors of these operators as bases for the signal representation, we develop topological signal processing tools for signals defined over CMCs. Then, we focus on the signal spectral representation and on the filtering of noisy flows observed over the cross-edges between different layers of CMCs. We show that a local signal representation
Traditional graph signal processing (GSP) methods applied to brain networks focus on signals defined on the nodes. Thus, they are unable to capture potentially important dynamics occurring on the edges. In this work, we adopt an edge-centric GSP approach to analyze edge signals constructed from 100 unrelated subjects of the Human Connectome Project. Specifically, we describe structural connectivity through the lens of the 1-dimensional Hodge Laplacian, processing signals defined on edges to capture co-fluctuation information between brain regions. We demonstrate that edge-based approaches achieve superior task decoding accuracy in static and dynamic scenarios compared to conventional node-based techniques, thereby unveiling unique aspects of brain functional organization. These findings underscore the promise of edge-focused GSP strategies for deepening our understanding of brain connectivity and functional dynamics.
The study of the interactions among different types of interconnected systems in complex networks has attracted significant interest across many research fields. However, effective signal processing over layered networks requires topological descriptors of the intra- and cross-layers relationships that are able to disentangle the homologies of different domains, at different scales, according to the specific learning task. In this paper, we present Cell MultiComplex (CMC) spaces, which are novel topological domains for representing multiple higher-order relationships among interconnected complexes. We introduce cross-Laplacians matrices, which are algebraic descriptors of CMCs enabling the extraction of topological invariants at different scales, whether global or local, inter-layer or intra-layer. Using the eigenvectors of these cross-Laplacians as signal bases, we develop topological signal processing tools for CMC spaces. In this first study, we focus on the representation and filtering of noisy flows observed over cross-edges between different layers of CMCs to identify cross-layer hubs, i.e., key nodes on one layer controlling the others.
The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and then it is especially useful to deal with signals defined over non-metric spaces. We focus on signals defined over simplicial complexes. Graph Signal Processing (GSP) represents a special case of Topological Signal Processing (TSP), referring to the situation where the signals are associated only with the vertices of a graph. Even though the theory can be applied to signals of any order, we focus on signals defined over the edges of a graph and show how building a simplicial complex of order two, i.e. including triangles, yields benefits in the analysis of edge signals. After reviewing the basic principles of algebraic topology, we derive a sampling theory for signals of any order and emphasize the interplay between signals of different order. Then we propose a method to infer the topology of a simplicial complex from data. We conclude with applications to real edge signals and to the analysis of discrete vector fields to illustrate the benefits of
Multi-channel acoustic signal processing is a well-established and powerful tool to exploit the spatial diversity between a target signal and non-target or noise sources for signal enhancement. However, the textbook solutions for optimal data-dependent spatial filtering rest on the knowledge of second-order statistical moments of the signals, which have traditionally been difficult to acquire. In this contribution, we compare model-based, purely data-driven, and hybrid approaches to parameter estimation and filtering, where the latter tries to combine the benefits of model-based signal processing and data-driven deep learning to overcome their individual deficiencies. We illustrate the underlying design principles with examples from noise reduction, source separation, and dereverberation.
This monograph presents a theoretical background and a broad introduction to the Min-Max Framework for Majorization-Minimization (MM4MM), an algorithmic methodology for solving minimization problems by formulating them as min-max problems and then employing majorization-minimization. The monograph lays out the mathematical basis of the approach used to reformulate a minimization problem as a min-max problem. With the prerequisites covered, including multiple illustrations of the formulations for convex and non-convex functions, this work serves as a guide for developing MM4MM-based algorithms for solving non-convex optimization problems in various areas of signal processing. As special cases, we discuss using the majorization-minimization technique to solve min-max problems encountered in signal processing applications and min-max problems formulated using the Lagrangian. Lastly, we present detailed examples of using MM4MM in ten signal processing applications such as phase retrieval, source localization, independent vector analysis, beamforming and optimal sensor placement in wireless sensor networks. The devised MM4MM algorithms are free of hyper-parameters and enjoy the advantag
Advances in machine learning technology have enabled real-time extraction of semantic information in signals which can revolutionize signal processing techniques and improve their performance significantly for the next generation of applications. With the objective of a concrete representation and efficient processing of the semantic information, we propose and demonstrate a formal graph-based semantic language and a goal filtering method that enables goal-oriented signal processing. The proposed semantic signal processing framework can easily be tailored for specific applications and goals in a diverse range of signal processing applications. To illustrate its wide range of applicability, we investigate several use cases and provide details on how the proposed goal-oriented semantic signal processing framework can be customized. We also investigate and propose techniques for communications where sensor data is semantically processed and semantic information is exchanged across a sensor network.
Brain foundation models (BFMs) have emerged as a transformative paradigm in computational neuroscience, offering a revolutionary framework for processing diverse neural signals across different brain-related tasks. These models leverage large-scale pre-training techniques, allowing them to generalize effectively across multiple scenarios, tasks, and modalities, thus overcoming the traditional limitations faced by conventional artificial intelligence (AI) approaches in understanding complex brain data. By tapping into the power of pretrained models, BFMs provide a means to process neural data in a more unified manner, enabling advanced analysis and discovery in the field of neuroscience. In this survey, we define BFMs for the first time, providing a clear and concise framework for constructing and utilizing these models in various applications. We also examine the key principles and methodologies for developing these models, shedding light on how they transform the landscape of neural signal processing. This survey presents a comprehensive review of the latest advancements in BFMs, covering the most recent methodological innovations, novel views of application areas, and challenges
The term "differentiable digital signal processing" describes a family of techniques in which loss function gradients are backpropagated through digital signal processors, facilitating their integration into neural networks. This article surveys the literature on differentiable audio signal processing, focusing on its use in music & speech synthesis. We catalogue applications to tasks including music performance rendering, sound matching, and voice transformation, discussing the motivations for and implications of the use of this methodology. This is accompanied by an overview of digital signal processing operations that have been implemented differentiably. Finally, we highlight open challenges, including optimisation pathologies, robustness to real-world conditions, and design trade-offs, and discuss directions for future research.
Water management is one of the most critical aspects of our society, together with population increase and climate change. Water scarcity requires a better characterization and monitoring of Water Distribution Networks (WDNs). This paper presents a novel framework for monitoring Water Distribution Networks (WDNs) by integrating physics-informed modeling of the nonlinear interactions between pressure and flow data with Topological Signal Processing (TSP) techniques. We represent pressure and flow data as signals defined over a second-order cell complex, enabling accurate estimation of water pressures and flows throughout the entire network from sparse sensor measurements. By formalizing hydraulic conservation laws through the TSP framework, we provide a comprehensive representation of nodal pressures and edge flows that incorporate higher-order interactions captured through the formalism of cell complexes. This provides a principled way to decompose the water flows in WDNs in three orthogonal signal components (irrotational, solenoidal and harmonic). The spectral representations of these components inherently reflect the conservation laws governing the water pressures and flows. Spa
In graph signal processing (GSP), prior information on the dependencies in the signal is collected in a graph which is then used when processing or analyzing the signal. Blind source separation (BSS) techniques have been developed and analyzed in different domains, but for graph signals the research on BSS is still in its infancy. In this paper, this gap is filled with two contributions. First, a nonparametric BSS method, which is relevant to the GSP framework, is refined, the Cramér-Rao bound (CRB) for mixing and unmixing matrix estimators in the case of Gaussian moving average graph signals is derived, and for studying the achievability of the CRB, a new parametric method for BSS of Gaussian moving average graph signals is introduced. Second, we also consider BSS of non-Gaussian graph signals and two methods are proposed. Identifiability conditions show that utilizing both graph structure and non-Gaussianity provides a more robust approach than methods which are based on only either graph dependencies or non-Gaussianity. It is also demonstrated by numerical study that the proposed methods are more efficient in separating non-Gaussian graph signals.
Powerful artificial intelligence (AI) tools that have emerged in recent years -- including large language models, automated coding assistants, and advanced image and speech generation technologies -- are the result of monumental human achievements. These breakthroughs reflect mastery across multiple technical disciplines and the resolution of significant technological challenges. However, some of the most profound challenges may still lie ahead. These challenges are not purely technical but pertain to the fair and responsible use of AI in ways that genuinely improve the global human condition. This article explores one promising application aligned with that vision: the use of AI tools to facilitate and enhance education, with a specific focus on signal processing (SP). It presents two interrelated perspectives: identifying and addressing technical limitations, and applying AI tools in practice to improve educational experiences. Primers are provided on several core technical issues that arise when using AI in educational settings, including how to ensure fairness and inclusivity, handle hallucinated outputs, and achieve efficient use of resources. These and other considerations --
After nearly a century of specialized applications in optics, remote sensing, and acoustics, the near-field (NF) electromagnetic propagation zone is experiencing a resurgence in research interest. This renewed attention is fueled by the emergence of promising applications in various fields such as wireless communications, holography, medical imaging, and quantum-inspired systems. Signal processing within NF sensing and wireless communications environments entails addressing issues related to extended scatterers, range-dependent beampatterns, spherical wavefronts, mutual coupling effects, and the presence of both reactive and radiative fields. Recent investigations have focused on these aspects in the context of extremely large arrays and wide bandwidths, giving rise to novel challenges in channel estimation, beamforming, beam training, sensing, and localization. While NF optics has a longstanding history, advancements in NF phase retrieval techniques and their applications have lately garnered significant research attention. Similarly, utilizing NF localization with acoustic arrays represents a contemporary extension of established principles in NF acoustic array signal processing.