Interpolation of Sparse Graph Signals by Sequential Adaptive Thresholds

This paper considers the problem of interpolating signals defined on graphs. A major presumption considered by many previous approaches to this problem has been lowpass/ band-limitedness of the underlying graph signal. However, inspired by the findings on sparse signal reconstruction, we consider the graph signal to be rather sparse/compressible in the Graph Fourier Transform (GFT) domain and propose the Iterative Method with Adaptive Thresholding for Graph Interpolation (IMATGI) algorithm for sparsity promoting interpolation of the underlying graph signal.We analytically prove convergence of the proposed algorithm. We also demonstrate efficient performance of the proposed IMATGI algorithm in reconstructing randomly generated sparse graph signals. Finally, we consider the widely desirable application of recommendation systems and show by simulations that IMATGI outperforms state-of-the-art algorithms on the benchmark datasets in this application.Comment: 12th International Conference on Sampling Theory and Applications (SAMPTA 2017

Random sampling of bandlimited signals on graphs

We study the problem of sampling k-bandlimited signals on graphs. We propose two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure, and its performance depends on a parameter called the graph coherence. On the contrary, the second strategy is adaptive but yields optimal results. Indeed, no more than O(k log(k)) measurements are sufficient to ensure an accurate and stable recovery of all k-bandlimited signals. This second strategy is based on a careful choice of the sampling distribution, which can be estimated quickly. Then, we propose a computationally efficient decoder to reconstruct k-bandlimited signals from their samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we conduct several experiments to test these techniques

Structured sampling and fast reconstruction of smooth graph signals

This work concerns sampling of smooth signals on arbitrary graphs. We first study a structured sampling strategy for such smooth graph signals that consists of a random selection of few pre-defined groups of nodes. The number of groups to sample to stably embed the set of $k$-bandlimited signals is driven by a quantity called the \emph{group} graph cumulative coherence. For some optimised sampling distributions, we show that sampling $O(k\log(k))$ groups is always sufficient to stably embed the set of $k$-bandlimited signals but that this number can be smaller -- down to $O(\log(k))$ -- depending on the structure of the groups of nodes. Fast methods to approximate these sampling distributions are detailed. Second, we consider $k$-bandlimited signals that are nearly piecewise constant over pre-defined groups of nodes. We show that it is possible to speed up the reconstruction of such signals by reducing drastically the dimension of the vectors to reconstruct. When combined with the proposed structured sampling procedure, we prove that the method provides stable and accurate reconstruction of the original signal. Finally, we present numerical experiments that illustrate our theoretical results and, as an example, show how to combine these methods for interactive object segmentation in an image using superpixels

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

A Spectral Graph Uncertainty Principle

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral "spreads" are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within $\varepsilon$ by a fast approximation algorithm requiring $O(\varepsilon^{-1/2})$ typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.Comment: 40 pages, 8 figure

Apprentissage automatique profond pour la modélisation de sous-maille en simulations aux grandes échelles de combustion prémélangée turbulente

Dans un siècle défini par le changement climatique et l'abondance de données, la combustion se dirige vers de nouvelles opportunités créées par la révolution numérique. Les simulations aux grandes échelles (Large Eddy Simulations, LES) de systèmes de combustion à échelle réelle deviennent réalisables, mais leur capacité prédictive se base sur la précision de modèles de sous-maille (Subgrid-Scale, SGS) qui tiennent compte de l'activité de combustion turbulent non résolue. L'apprentissage automatique profond (Deep Learning, DL) a récemment été utilisé pour entraîner des modèles SGS basés sur les données qui atteignent une excellente précision lors de tests a priori. Toutefois, il n'y a toujours presque pas d'applications de modèles DL SGS à des LES de systèmes de combustion industriels. Ces travaux s'intéressent à trois éléments qui doivent être étudiés pour permettre l'adoption du DL dans des LES de combustion turbulente prémélangée : l'évaluation de modèles DL sur des cas tests à haut Reynolds, l'assurance de leur capacité à généraliser au-delà de leur configuration d'entraînement, et l'implémentation d'une intégration efficace de modèles DL à des solveurs LES haute performance. Trois modèles DL incluant graduellement chacun de ces éléments sont développés. Ils sont basés sur des réseaux de neurones convolutionnels (Convolutional Neural Networks, CNNs) U-Nets entraînés sur des instantanés filtrés et déraffinés de simulations numériques directes. Premièrement, un modèle pour la densité totale de surface de flamme est entraîné sur la flamme de jet turbulente à haut Reynolds R2. Une excellente généralisation a priori à de plus hauts nombres de Reynolds et à des instantanés LES est observée, et des aperçus sur le fonctionnement interne du modèle sont proposés. Dans un second temps, un modèle CNN pour la variance SGS de la variable de progrès est entraîné sur une flamme plane turbulente statistiquement stationnaire. Avec une formulation Pfitzner du terme source et une fermeture beta densité de probabilité, il est capable de prédire a priori avec précision la variance SGS et le taux de réaction filtré sur la flamme de jet R2, démontrant ainsi sa capacité à généraliser à de nouvelles configurations. Troisièmement, la stratégie de couplage AVBP-DL est développée pour permettre à des modèles DL d'être interrogés par le solveur AVBP avec un surcoût de calcul négligeable. Enfin, le cas test d'explosion aérée et obstruée Masri est utilisé pour tester a posteriori un modèle CNN pour le facteur de plissement SGS entraîné sur la flamme plane turbulente statistiquement stationnaire. Le modèle prédit la bonne suppression maximale, mais ceci résulte d'une compensation entre un plissement excessif lors de la phase initiale laminaire et d'un plissement insuffisant durant l'étape critique de propagation turbulente. Plusieurs tentatives de correction de ce comportement sont ensuite explorées

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described