Joint Graph Learning and Blind Separation of Smooth Graph Signals Using Minimization of Mutual Information and Laplacian Quadratic Forms

Abstract

The smoothness of graph signals has found desirable real applications for processing irregular (graph-based) signals. When the latent sources of the mixtures provided to us as observations are smooth graph signals, it is more efficient to use graph signal smoothness terms along with the classic independence criteria in Blind Source Separation (BSS) approaches. In the case of underlying graphs being known, Graph Signal Processing (GSP) provides valuable tools; however, in many real applications, these graphs can not be well-defined <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> and need to be learned from data. In this paper, a GSP-based approach for joint Graph Learning (GL) and BSS of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">smooth</i> graph signal sources is proposed, which, to the best of our knowledge, has not been addressed. The minimization of Mutual Information (MI) is exploited in our work because of its desirable advantages, such as having no local minimum and exact (not approximative) modeling of statistical independence. As well as investigating the convergence analysis and identifiability conditions and deriving Cramér-Rao Lower Bound (CRLB), comprehensive experiments of generating (cyclostationary) graph signal mixtures are provided to show the superiority over classic and also GSP-based BSS approaches in the case of the latent sources being <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">smooth</i> on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unknown</i> underlying graphs. Based on the provided experimental results, our proposed method performs quite robustly in even high noise levels in both terms of source separation and recovery of underlying graphs.