加权单纯形上的拓扑信号处理
Topological Signal Processing over Weighted Simplicial Complexes
摘要：
在许多信号处理和机器学习应用中，对可以表示和分析数据的拓扑域进行加权是一种关键策略，能够提取和利用有意义的数据特征及其（高阶）关系。本文的目标是提出加权单纯复形的拓扑信号处理工具。具体来说，依靠加权霍奇拉普拉斯理论，我们提出了有效的策略，共同学习复杂的权重和滤波器的螺线管，无旋和谐波分量的信号定义在复杂的。我们数值评估所提出的程序的有效性。
英文摘要：
Weighing the topological domain over which data can be represented and analysed is a key strategy in many signal processing and machine learning applications, enabling the extraction and exploitation of meaningful data features and their (higher order) relationships. Our goal in this paper is to present topological signal processing tools for weighted simplicial complexes. Specifically, relying on the weighted Hodge Laplacian theory, we propose efficient strategies to jointly learn the weights of the complex and the filters for the solenoidal, irrotational and harmonic components of the signals defined over the complex. We numerically asses the effectiveness of the proposed procedures.
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