A Unified Approach for Simultaneous Graph Learning and Blind Separation of Graph Signal Sources
Aref Einizade, Sepideh Hajipour Sardouie · IEEE Transactions on Signal and Information Processing over Networks · 2022
Blind separation of the sources supported by graphs, i.e., graph signals, is a nascent and challenging Blind Source Separation (BSS) problem. In these cases, along with the statistical independence of the sources, additional dependency information can be interpreted from their graph structure, which provides exploiting different Graph Signal Processing (GSP) tools to improve the separation performance. To the best of our knowledge, in these cases, only GraDe and GraphJADE methods have been proposed to exploit the graph dependencies and/or GSP techniques to improve the separation quality rather than the conventional BSS methods. Despite the significant advantages of these graph-based methods, assuming that the underlying graphs are known is a serious drawback, especially in many real-world applications. To address this issue, in this paper, we propose a Unified objective function for GraphJADE with Graph Learning, namely U-GraphJADE-GL, and use the Block Coordinate Descent (BCD) to optimize it. In our proposed method, along with the separation task, the underlying graphs are learned simultaneously in an efficient manner. We compare the performance of the U-GraphJADE-GL with that of the GraDe with Graph Learning (U-GraDe-GL) and the conventional BSS methods in the BSS task and also analyze the GL performance. Besides, as well as the theoretical and experimental convergence analysis, we derive/state the Cramér-Rao bound (CRB) on the estimation of the mixing and unmixing matrices and also on the attainable Interference-to-Source Ratio (ISR) and compare the asymptotic performance of the proposed method with that of the optimal CRB estimators. To investigate the applicability in real applications, the performance of the proposed method is also compared with that of the abovementioned algorithms for denoising the epileptic Electroencephalogram (EEG) signals and also for the audio speech source separation task. The results show the superiority of the proposed algorithm compared to the other methods in the case of BSS of graph signals with unknown graphs.