On the Graph Fourier Transform for Directed Graphs

The analysis of signals defined over a graph is relevant in many applications, such as social and economic networks, big data or biological networks, and so on. A key tool for analyzing these signals is the so-called graph Fourier transform (GFT). Alternative definitions of GFT have been suggested i...

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Bibliographic Details
Published inIEEE journal of selected topics in signal processing Vol. 11; no. 6; pp. 796 - 811
Main Authors Sardellitti, Stefania, Barbarossa, Sergio, Di Lorenzo, Paolo
Format Journal Article
LanguageEnglish
Published IEEE 01.09.2017
Subjects
Clustering
Fourier transforms
graph Fourier transform
graph signal processing
Laplace equations
Linear programming
Matrix decomposition
Minimization
Signal processing
total variation
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Summary:The analysis of signals defined over a graph is relevant in many applications, such as social and economic networks, big data or biological networks, and so on. A key tool for analyzing these signals is the so-called graph Fourier transform (GFT). Alternative definitions of GFT have been suggested in the literature, based on the eigen-decomposition of either the graph Laplacian or adjacency matrix. In this paper, we address the general case of directed graphs and we propose an alternative approach that builds the graph Fourier basis as the set of orthonormal vectors that minimize a continuous extension of the graph cut size, known as the Lovász extension. To cope with the nonconvexity of the problem, we propose two alternative iterative optimization methods, properly devised for handling orthogonality constraints. Finally, we extend the method to minimize a continuous relaxation of the balanced cut size. The formulated problem is again nonconvex, and we propose an efficient solution method based on an explicit-implicit gradient algorithm.
ISSN:1932-4553
1941-0484
DOI:10.1109/JSTSP.2017.2726979