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A Graph Signal Processing solution for Defective Directed Graphs

Michael Shell,  John Doe,  and Jane Doe M. Shell was with the Department of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332 USA e-mail: (see http://www.michaelshell.org/contact.html).J. Doe and J. Doe are with Anonymous University.Manuscript received April 19, 2005; revised August 26, 2015.
Abstract

The main purpose of this thesis is to find a method that allows to systematically adapt graph signal processing (GSP) techniques so they can be used in every graph, and especially on graphs with a non-diagonalizable graph operator. We begin by presenting the framework in which GSP is developed and studying why the computation of the Graph Fourier Transform (GFT) is problematic for defective directed graphs. We introduce our proposed method, which can be used to form, based on the spectral decomposition of a matrix obtained through its Schur decomposition, a complete basis of vectors that can be used as a replacement of the previously mentioned GFT. The proposed method, the Graph Schur Transform (GST), aims to offer a valid operator to perform a spectral decomposition that can be used even in the case of defective matrices. Finally, we study the main properties of the proposed method and compare them with the corresponding properties offered by the Diffusion Wavelets design and we prove, for a large set of directed graphs, that the GST provides a valid solution for the problem.

Index Terms:
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Appendix A Proof of the First Zonklar Equation

Appendix one text goes here.

Appendix B

Appendix two text goes here.

Acknowledgment

The authors would like to thank…

References