Analyse harmonique sur graphes dirigés et applications : de l'analyse de Fourier aux ondelettes

Harry Sevi 1, 2
Abstract : The research conducted in this thesis aims to develop a harmonic analysis for functions defined on the vertices of an oriented graph. In the era of data deluge, much data is in the form of graphs and data on this graph. In order to analyze and exploit this graph data, we need to develop mathematical and numerically efficient methods. This development has led to the emergence of a new theoretical framework called signal processing on graphs, which aims to extend the fundamental concepts of conventional signal processing to graphs. Inspired by the multi-scale aspect of graphs and graph data, many multi-scale constructions have been proposed. However, they apply only to the non-directed framework. The extension of a harmonic analysis on an oriented graph, although natural, is complex. We, therefore, propose a harmonic analysis using the random walk operator as the starting point for our framework. First, we propose Fourier-type bases formed by the eigenvectors of the random walk operator. From these Fourier bases, we determine a frequency notion by analyzing the variation of its eigenvectors. The determination of a frequency analysis from the basis of the vectors of the random walk operator leads us to multi-scale constructions on oriented graphs. More specifically, we propose a wavelet frame construction as well as a decimated wavelet construction on directed graphs. We illustrate our harmonic analysis with various examples to show its efficiency and relevance.
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Harry Sevi. Analyse harmonique sur graphes dirigés et applications : de l'analyse de Fourier aux ondelettes. Analyse fonctionnelle [math.FA]. Université de Lyon, 2018. Français. ⟨NNT : 2018LYSEN068⟩. ⟨tel-02024654⟩

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