A graph signal processing solution for defective directed graphs

Abstract

The main purpose of this thesis is to nd a method that allows to systematically adapt GSP techniques so they can be used on most non-diagonalizable graph operators. In Chapter 1 we begin by presenting the framework in which GSP is developed, giving some basic de nitions in the eld of graph theory and in relation with graph signals. We also present the concept of a Graph Fourier Tranform (GFT), which will be of great importance in the proposed solution. Chapter 2 presents the actual motivation of the research: Why the computation of the GFT is problematic for some directed graphs, and the speci c cases in which this happen. We will see that the issue can not be assigned to a very speci c graph topography, and therefore it is important to develop solutions that can be applied to any directed graph. In Chapter 3 we introduce our proposed new 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 Graph Fourier Transform. The proposed method, the Graph Schur Transform (GST), aims to o er a valid operator to perform a spectral decomposition of a graph that can be used even in the case of defective matrices. Finally, in Chapter 4 we study the main properties of the proposed method and compare them with the corresponding properties o ered by the Di usion Wavelets design. In the last section we prove, for a large set of directed graphs, that the GST provides a valid solution for the problem