Benjamin Ricaud

A windowed graph Fourier transform

The prevalence of signals on weighted graphs is increasing; however, because of the irregular structure of weighted graphs, classical signal processing techniques cannot be directly applied to signals on graphs. In this paper, we define generalized translation and modulation operators for signals on graphs, and use these operators to adapt the classical windowed Fourier transform to the graph setting, enabling vertex-frequency analysis. When we apply this transform to a signal with frequency components that vary along a path graph, the resulting spectrogram matches our intuition from classical discrete-time signal processing. Yet, our construction is fully generalized and can be applied to analyze signals on any undirected, connected, weighted graph.

A survey of uncertainty principles and some signal processing applications

The goal of this paper is to review the main trends in the domain of uncertainty principles and localization, highlight their mutual connections and investigate practical consequences. The discussion is strongly oriented towards, and motivated by signal processing problems, from which significant advances have been made recently. Relations with sparse approximation and coding problems are emphasized.

Principal Patterns on Graphs: Discovering Coherent Structures in Datasets

Graphs are now ubiquitous in almost every field of research. Recently, new research areas devoted to the analysis of graphs and data associated to their vertices have emerged. Focusing on dynamical processes, we propose a fast, robust, and scalable framework for retrieving and analyzing recurring patterns of activity on graphs. Our method relies on a novel type of multilayer graph that encodes the spreading or propagation of events between successive time steps. We demonstrate the versatility of our method by applying it on three different real-world examples. First, we study how rumor spreads on a social network. Second, we reveal congestion patterns of pedestrians in a train station. Finally, we show how patterns of audio playlists can be used in a recommender system. In each example, relevant information previously hidden in the data is extracted in a very efficient manner, emphasizing the scalability of our method. With a parallel implementation scaling linearly with the size of the dataset, our framework easily handles millions of nodes on a single commodity server.

An optimally concentrated Gabor transform for localized time-frequency components

Gabor analysis is one of the most common instances of time-frequency signal analysis. Choosing a suitable window for the Gabor transform of a signal is often a challenge for practical applications, in particular in audio signal processing. Many time-frequency (TF) patterns of different shapes may be present in a signal and they can not all be sparsely represented in the same spectrogram. We propose several algorithms, which provide optimal windows for a user-selected TF pattern with respect to different concentration criteria. We base our optimization algorithm on lp-norms as measure of TF spreading. For a given number of sampling points in the TF plane we also propose optimal lattices to be used with the obtained windows. We illustrate the potentiality of the method on selected numerical examples.

Refined Support and Entropic Uncertainty Inequalities

Generalized versions of the entropic (Hirschman- Beckner) and support (Elad-Bruckstein) uncertainty principle are presented for frames representations. Moreover, a sharpened version of the support inequality is obtained by introducing a generalization of the coherence. In the finite-dimensional case and under certain conditions, minimizers of these inequalities are given. In addition, lp norms inequalities are introduced as byproducts of the entropic inequalities.

Effective models for excitons in carbon nanotubes

Abstract.We analyse the low lying spectrum of a model of excitons in carbon nanotubes. Consider two particles with opposite charges and a Coulomb self-interaction, placed on an infinitely long cylinder. If the cylinder radius becomes small, the low lying spectrum of their relative motion is well described by a one-dimensional effective Hamiltonian which is exactly solvable.

On the Sparsity of Wavelet Coefficients for Signals on Graphs

A number of new localized, multiscale transforms have recently been introduced to analyze data residing on weighted graphs. In signal processing tasks such as regularization and compression, much of the power of classical wavelets on the real line is derived from their theoretically and empirically proven ability to sparsely represent piecewise-smooth signals, which appear to be locally polynomial at sufficiently small scales. As of yet in the graph setting, there is little mathematical theory relating the sparsity of localized, multiscale transform coefficients to the structures of graph signals and their underlying graphs. In this paper, we begin to explore notions of global and local regularity of graph signals, and analyze the decay of spectral graph wavelet coefficients for regular graph signals.

Tracking time-vertex propagation using dynamic graph wavelets

Graph Signal Processing generalizes classical signal processing to signal or data indexed by the vertices of a weighted graph. So far, the research efforts have been focused on static graph signals. However numerous applications involve graph signals evolving in time, such as spreading or propagation of waves on a network. The analysis of this type of data requires a new set of methods that takes into account the time and graph dimensions. We propose a novel class of wavelet frames named Dynamic Graph Wavelets, whose time-vertex evolution follows a dynamic process. We demonstrate that this set of functions can be combined with sparsity based approaches such as compressive sensing to reveal information on the dynamic processes occurring on a graph. Experiments on real seismological data show the efficiency of the technique, allowing to estimate the epicenter of earthquake events recorded by a seismic network.

Locating Temporal Functional Dynamics of Visual Short-Term Memory Binding using Graph Modular Dirichlet Energy

Visual short-term memory binding tasks are a promising early biomarker for Alzheimers disease (AD). We probe the transient physiological underpinnings of these tasks over the healthy brains functional connectome by contrasting shape only (Shape) and shape-colour binding (Bind) conditions, displayed in the left and right sides of the screen, separately, in young volunteers. Electroencephalogram recordings during the encoding and maintenance periods of these tasks are analysed using functional connectomics. Particularly, we introduce and implement a novel technique named Modular Dirichlet Energy (MDE) which allows robust and flexible analysis of the connectome with unprecedentedly high temporal precision. We find that connectivity in the Bind condition is stronger than in the Shape condition in both occipital and frontal network modules during the encoding period of the right screen condition but not the left screen condition. Using MDE we are able to discern driving effects in the occipital module between 100-140ms, which noticeably coincides with the P100 visually evoked potential, and a driving effect in the interaction of occipital and frontal modules between 120-140ms, suggesting a delayed information processing difference between these modules. This provides temporally precise information over a heterogenous population for tasks related to the sensitive and specific detection of AD.Visual short-term memory binding tasks are a promising early marker for Alzheimer’s disease (AD). To uncover functional deficits of AD in these tasks it is meaningful to first study unimpaired brain function. Electroencephalogram recordings were obtained from encoding and maintenance periods of tasks performed by healthy young volunteers. We probe the task’s transient physiological underpinnings by contrasting shape only (Shape) and shape-colour binding (Bind) conditions, displayed in the left and right sides of the screen, separately. Particularly, we introduce and implement a novel technique named Modular Dirichlet Energy (MDE) which allows robust and flexible analysis of the functional network with unprecedented temporal precision. We find that connectivity in the Bind condition is less integrated with the global network than in the Shape condition in occipital and frontal modules during the encoding period of the right screen condition. Using MDE we are able to discern driving effects in the occipital module between 100–140 ms, coinciding with the P100 visually evoked potential, followed by a driving effect in the frontal module between 140–180 ms, suggesting that the differences found constitute an information processing difference between these modules. This provides temporally precise information over a heterogeneous population in promising tasks for the detection of AD.

PCA using graph total variation

Mining useful clusters from high dimensional data has received significant attention of the signal processing and machine learning community in the recent years. Linear and non-linear dimensionality reduction has played an important role to overcome the curse of dimensionality. However, often such methods are accompanied with problems such as high computational complexity (usually associated with the nuclear norm minimization), non-convexity (for matrix factorization methods) or susceptibility to gross corruptions in the data. In this paper we propose a convex, robust, scalable and efficient Principal Component Analysis (PCA) based method to approximate the low-rank representation of high dimensional datasets via a two-way graph regularization scheme. Compared to the exact recovery methods, our method is approximate, in that it enforces a piecewise constant assumption on the samples using a graph total variation and a piecewise smoothness assumption on the features using a graph Tikhonov regularization. Futhermore, it retrieves the low-rank representation in a time that is linear in the number of data samples. Clustering experiments on 3 benchmark datasets with different types of corruptions show that our proposed model outperforms 7 state-of-the-art dimensionality reduction models.