In the nascent and challenging problem of the blind separation of the sources (BSS) supported by graphs, i.e., graph signals, along with the statistical independence of the sources, additional dependency information can be interpreted from their graph structure. To the best of our knowledge, in these cases, only GraDe and GraphJADE methods have been proposed to exploit the graph dependencies and/or Graph Signal Processing (GSP) techniques to improve the separation quality. Despite the significant advantages of these graph-based methods, they assume that the underlying graphs are known, which 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 (GL), namely U-GraphJADE-GL, and use the Block Coordinate Descent (BCD) to optimize it, which along with the separation task, the underlying graphs are learned simultaneously. We compare the performance of the U-GraphJADE-GL with the GraDe with GL (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 the optimal CRB estimators. To investigate the applicability in real applications, the proposed method is also successfully applied for denoising the epileptic Electroencephalogram (EEG) signals and also for the audio speech source separation task.

中文翻译：

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一种同时图学习和图信号源盲分离的统一方法

在由图（即图信号）支持的源盲分离（BSS）这一新生且具有挑战性的问题中，连同源的统计独立性，可以从它们的图结构中解释额外的依赖信息。据我们所知，在这些情况下，仅提出了 GraDe 和 GraphJADE 方法来利用图依赖性和/或图信号处理 (GSP) 技术来提高分离质量。尽管这些基于图的方法具有显着优势，但它们假设底层图是已知的，这是一个严重的缺点，尤其是在许多实际应用中。为了解决这个问题，在本文中，我们提出了一个带有图学习（GL）的 GraphJADE 统一目标函数，即 U-GraphJADE-GL，并使用块坐标下降（BCD）对其进行优化，与分离任务一起，同时学习底层图。我们在 BSS 任务中比较了 U-GraphJADE-GL 与 GraDe with GL (U-GraDe-GL) 和传统 BSS 方法的性能，并分析了 GL 性能。此外，除了理论和实验收敛性分析之外，我们还推导出/陈述了 Cramér-Rao 界 (CRB) 关于混合和解混合矩阵的估计以及可达到的干扰源比 (ISR)，并比较所提出的方法与最优 CRB 估计量的渐近性能。为了研究在实际应用中的适用性，