This paper presents a graph signal processing algorithm to uncover the intrinsic low-rank components and the underlying graph of a high-dimensional, graph-smooth and grossly-corrupted dataset. In our problem formulation, we assume that the perturbation on the low-rank components is sparse and the signal is smooth on the graph. We propose an algorithm to estimate the low-rank components with the help of the graph and refine the graph with better estimated low-rank components. We propose to perform the low-rank estimation and graph refinement jointly so that low-rank estimation can benefit from the refined graph, and graph refinement can leverage the improved low-rank estimation. We propose to address the problem with an alternating optimization. Moreover, we perform a mathematical analysis to understand and quantify the impact of the inexact graph on the low-rank estimation, justifying our scheme with graph refinement as an integrated step in estimating low-rank components. We perform extensive experiments on the proposed algorithm and compare with state-of-the-art low-rank estimation and graph learning techniques. Our experiments use synthetic data and real brain imaging (MEG) data that is recorded when subjects are presented with different categories of visual stimuli. We observe that our proposed algorithm is competitive in estimating the low-rank components, adequately capturing the intrinsic task-related information in the reduced dimensional representation, and leading to better performance in a classification task. Furthermore, we notice that our estimated graph indicates compatible brain active regions for visual activity as neuroscientific findings.

中文翻译：

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高维图信号中低秩分量和连通图的联合估计：在脑成像中的应用

本文提出了一种图形信号处理算法，以揭示高维、图形平滑和严重损坏的数据集的内在低阶组件和底层图形。在我们的问题表述中，我们假设低秩分量上的扰动是稀疏的，并且图上的信号是平滑的。我们提出了一种算法来在图的帮助下估计低秩分量，并用更好的估计低秩分量细化图。我们建议联合执行低秩估计和图细化，以便低秩估计可以从细化图中受益，而图细化可以利用改进的低秩估计。我们建议通过交替优化来解决这个问题。而且，我们进行数学分析以理解和量化不精确图对低秩估计的影响，证明我们的方案与图细化作为估计低秩分量的综合步骤的合理性。我们对所提出的算法进行了大量实验，并与最先进的低秩估计和图学习技术进行了比较。我们的实验使用合成数据和真实大脑成像 (MEG) 数据，这些数据是在受试者接受不同类别的视觉刺激时记录的。我们观察到我们提出的算法在估计低阶组件方面具有竞争力，在降维表示中充分捕获与任务相关的内在信息，并在分类任务中获得更好的性能。此外，