Robust Network Topology Inference and Processing of Graph Signals

The abundance of large and heterogeneous systems is rendering contemporary data more pervasive, intricate, and with a non-regular structure. With classical techniques facing troubles to deal with the irregular (non-Euclidean) domain where the signals are defined, a popular approach at the heart of graph signal processing (GSP) is to: (i) represent the underlying support via a graph and (ii) exploit the topology of this graph to process the signals at hand. In addition to the irregular structure of the signals, another critical limitation is that the observed data is prone to the presence of perturbations, which, in the context of GSP, may affect not only the observed signals but also the topology of the supporting graph. Ignoring the presence of perturbations, along with the couplings between the errors in the signal and the errors in their support, can drastically hinder estimation performance. While many GSP works have looked at the presence of perturbations in the signals, much fewer have looked at the presence of perturbations in the graph, and almost none at their joint effect. While this is not surprising (GSP is a relatively new field), we expect this to change in the upcoming years. Motivated by the previous discussion, the goal of this thesis is to advance toward a robust GSP paradigm where the algorithms are carefully designed to incorporate the influence of perturbations in the graph signals, the graph support, and both. To do so, we consider different types of perturbations, evaluate their disruptive impact on fundamental GSP tasks, and design robust algorithms to address them.Comment: Dissertatio

Distributed Adaptive Learning of Graph Signals

The aim of this paper is to propose distributed strategies for adaptive learning of signals defined over graphs. Assuming the graph signal to be bandlimited, the method enables distributed reconstruction, with guaranteed performance in terms of mean-square error, and tracking from a limited number of sampled observations taken from a subset of vertices. A detailed mean square analysis is carried out and illustrates the role played by the sampling strategy on the performance of the proposed method. Finally, some useful strategies for distributed selection of the sampling set are provided. Several numerical results validate our theoretical findings, and illustrate the performance of the proposed method for distributed adaptive learning of signals defined over graphs.Comment: To appear in IEEE Transactions on Signal Processing, 201

From spline wavelet to sampling theory on circulant graphs and beyond– conceiving sparsity in graph signal processing

Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs. Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations. Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes. Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.Open Acces

Applied Harmonic Analysis and Sparse Approximation

Efficiently analyzing functions, in particular multivariate functions, is a key problem in applied mathematics. The area of applied harmonic analysis has a significant impact on this problem by providing methodologies both for theoretical questions and for a wide range of applications in technology and science, such as image processing. Approximation theory, in particular the branch of the theory of sparse approximations, is closely intertwined with this area with a lot of recent exciting developments in the intersection of both. Research topics typically also involve related areas such as convex optimization, probability theory, and Banach space geometry. The workshop was the continuation of a first event in 2012 and intended to bring together world leading experts in these areas, to report on recent developments, and to foster new developments and collaborations

Sensing physical fields: Inverse problems for the diffusion equation and beyond

Due to significant advances made over the last few decades in the areas of (wireless) networking, communications and microprocessor fabrication, the use of sensor networks to observe physical phenomena is rapidly becoming commonplace. Over this period, many aspects of sensor networks have been explored, yet a thorough understanding of how to analyse and process the vast amounts of sensor data collected, remains an open area of research. This work therefore, aims to provide theoretical, as well as practical, advances this area. In particular, we consider the problem of inferring certain underlying properties of the monitored phenomena, from our sensor measurements. Within mathematics, this is commonly formulated as an inverse problem; whereas in signal processing it appears as a (multidimensional) sampling and reconstruction problem. Indeed it is well known that inverse problems are notoriously ill-posed and very demanding to solve; meanwhile viewing it as the latter also presents several technical challenges. In particular, the monitored field is usually nonbandlimited, the sensor placement is typically non-regular and the space-time dimensions of the field are generally non-homogeneous. Furthermore, although sensor production is a very advanced domain, it is near impossible and/or extremely costly to design sensors with no measurement noise. These challenges therefore motivate the need for a stable, noise robust, yet simple sampling theory for the problem at hand. In our work, we narrow the gap between the domains of inverse problems and modern sampling theory, and in so doing, extend existing results by introducing a framework for solving the inverse source problems for a class of some well-known physical phenomena. Some examples include: the reconstruction of plume sources, thermal monitoring of multi-core processors and acoustic source estimation, to name a few. We assume these phenomena and their sources can be described using partial differential equation (PDE) and parametric source models, respectively. Under this assumption, we obtain a well-posed inverse problem. Initially, we consider a phenomena governed by the two-dimensional diffusion equation -- i.e. 2-D diffusion fields, and assume that we have access to its continuous field measurements. In this setup, we derive novel exact closed-form inverse formulae that solve the inverse diffusion source problem, for a class of localized and non-localized source models. In our derivation, we prove that a particular 1-D sequence of, so called, generalized measurements of the field is governed by a power-sum series, hence it can be efficiently solved using existing algebraic methods such as Prony's method. Next, we show how to obtain these generalized measurements, by using Green's second identity to combine the continuous diffusion field with a family of well-chosen sensing functions. From these new inverse formulae, we therefore develop novel noise robust centralized and distributed reconstruction methods for diffusion fields. Specifically, we extend these inverse formulae to centralized sensor networks using numerical quadrature; conversely for distributed networks, we propose a new physics-driven consensus scheme to approximate the generalized measurements through localized interactions between the sensor nodes. Finally we provide numerical results using both synthetic and real data to validate the proposed algorithms. Given the insights gained, we eventually turn to the more general problem. That is, the two- and three-dimensional inverse source problems for any linear PDE with constant coefficients. Extending the previous framework, we solve the new class of inverse problems by establishing an otherwise subtle link with modern sampling theory. We achieved this by showing that, the desired generalized measurements can be computed by taking linear weighted-sums of the sensor measurements. The advantage of this is two-fold. First, we obtain a more flexible framework that permits the use of more general sensing functions, this freedom is important for solving the 3-D problem. Second, and remarkably, we are able to analyse many more physical phenomena beyond diffusion fields. We prove that computing the proper sequence of generalized measurements for any such field, via linear sums, reduces to approximating (a family of) exponentials with translates of a particular prototype function. We show that this prototype function depends on the Green's function of the field, and then derive an explicit formula to evaluate the proper weights. Furthermore, since we now have more freedom in selecting the sensing functions, we discuss how to make the correct choice whilst emphasizing how to retrieve the unknown source parameters from the resulting (multidimensional) Prony-like systems. Based on this new theory we develop practical, noise robust, sensor network strategies for solving the inverse source problem, and then present numerical simulation results to verify the performance of our proposed schemes.Open Acces