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Signal processing on simplicial complexes
arXiv - CS - Social and Information Networks Pub Date : 2021-06-14 , DOI: arxiv-2106.07471
Michael T. Schaub, Jean-Baptiste Seby, Florian Frantzen, T. Mitchell Roddenberry, Yu Zhu, Santiago Segarra

Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i.e., the higher-order or multi-way relations connecting the constituent entities. More recently, a number of studies have considered dynamical processes that explicitly account for such higher-order dependencies, e.g., in the context of epidemic spreading processes or opinion formation. In this chapter, we focus on a closely related, but distinct third perspective: how can we use higher-order relationships to process signals and data supported on higher-order network structures. In particular, we survey how ideas from signal processing of data supported on regular domains, such as time series or images, can be extended to graphs and simplicial complexes. We discuss Fourier analysis, signal denoising, signal interpolation, and nonlinear processing through neural networks based on simplicial complexes. Key to our developments is the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing.

中文翻译:

单纯复形的信号处理

迄今为止,高阶网络主要是在研究复杂系统结构的背景下考虑的,即连接组成实体的高阶或多向关系。最近,许多研究已经考虑了动态过程,这些过程明确解释了这种高阶依赖性,例如,在流行病传播过程或意见形成的背景下。在本章中,我们关注密切相关但又截然不同的第三个视角:我们如何使用高阶关系来处理高阶网络结构支持的信号和数据。特别是,我们调查了来自常规域(例如时间序列或图像)支持的数据的信号处理的想法如何扩展到图形和单纯复形。我们讨论傅立叶分析、信号去噪、信号插值、通过基于单纯复形的神经网络进行非线性处理。我们开发的关键是霍奇拉普拉斯矩阵,这是一种多关系算子,它利用单纯复形的特殊结构并概括了拉普拉斯矩阵在图信号处理中的理想属性。
更新日期:2021-06-15
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