![]() ![]() Various wavelet transforms can also be defined on graphs, and the teams involved in this proposal have pioneered the design of wavelet transforms on graphs using spectral graph theory. Thanks to spectral graph theory, a Fourier transform can be defined on graphs from the eigen decomposition of the graph’s Laplacian operator. Among signal processing tools, transforms naturally play a key role in modelling data. This has opened the path to many exciting future research, calling to revisit most of the usual signal processing tasks (filtering, denoising, compression, etc.). As such, “signal processing on graphs” (SPG) is an emerging topic, that has already lead to pioneering theoretical and practical work to formalize foundational definitions and tools. This algorithm is particularly efficient when the data structure is very irregular.Nowadays, more and more data natively “live” on the vertices of a graph: brain activity supported by neurons in networks, traffic on transport and energy networks, data from users of social media, complex 3D surfaces describing real objects… Although graphs have been extensively studied in mathematics and computer science, a “signal processing” viewpoint on these objects remains largely to be invented. Based on Lanczos’ polynomial approximation, it allows us to accurately and efficiently apply any filtering operations on graphs. To use it on big datasets and overcome the curse of dimensionality, we have developed a novel fast-filtering algorithm. Fast filtering algorithm: Graph filtering is the base of many algorithms. ![]() Large scale graph learning: Leveraging convex optimization and graph signal processing, we show how to scale the graph learning problem.We have been developing the basic tools for the analysis of such signals through multiple contributions. Graph signal prossessing of time series: Graphs are very convenient tools to model the structure behind multivariate time series.Available in MATLAB and Python, it is fully documented and provides a tutorial and some demonstration files. It contains efficient implementations of graph filtering operations that allow a rapid computation of the wavelet or Gabor transform. Graph signal processing toolbox: The graph signal processing toolbox is an open-source collection of algorithms combining my research with that of my co-workers.DeepSphere: Leveraging graph convolution neural networks, we have defined a new deep neural network architecture designed for spherical data.Instead of a single number, our uncertainty bound is a function evolving on the graph providing insights of the graph structure. Local uncertainty principle: The structure of the graph being far from regular, we propose the new concept of local uncertainty principle.This model is able to exploit the intrinsic links between the features and show very exciting results. Data probabilistic model: Generalizing the concept of stationarity on graphs, we propose a new model for data.The range of applications cannot be enumerated but contains, among others, automatic text classification, Internet page ranking, digit classification and movie recommendations. Almost all signal-processing tools have been generalized to graphs, thus allowing them to be used on irregular data structures. Common tasks in signal-processing on graphs include filtering, de-noising, in-painting, compression, clustering, partitioning, sparsification, features extraction, classification and regression. The graphs are usually used to discretize the abstract domain of the data and build accurate representations of them. Nevertheless, most of the time this does not happen and the graph is constructed directly from the data or from additional information. Social networks are one of the most common examples of such data since the graph is simply provided by the users themselves. Graph signal processing is a special area in signal processing based on spectral graph theory where the data possess an intrinsic graph structure.
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