单纯复形上的信号处理
Signal processing on simplicial complexes
摘要：
图形信号处理（GSP）的理论发展和应用已引起人们的广泛关注。在经典的GSP中，底层结构在维数方面受到限制。图是一种对二元关系进行建模的组合对象，它不直接对复杂的n元关系进行建模。图的一个可能的高维推广是单纯复形。它们是图的约束情形和超图的一般情形之间的一步。在本文中，我们开发了一个信号处理框架上的单纯复形，使我们恢复传统的GSP理论时，仅限于图上的信号。值得一提的是，该框架的工作更普遍，虽然本文的重点是单纯复形。我们演示了如何执行信号处理的框架，使用数值例子。
英文摘要：
Theoretical development and applications of graph signal processing (GSP) have attracted much attention. In classical GSP, the underlying structures are restricted in terms of dimensionality. A graph is a combinatorial object that models binary relations, and it does not directly model complex n-ary relations. One possible high dimensional generalization of graphs are simplicial complexes. They are a step between the constrained case of graphs and the general case of hypergraphs. In this paper, we develop a signal processing framework on simplicial complexes, such that we recover the traditional GSP theory when restricted to signals on graphs. It is worth mentioning that the framework works much more generally, though the focus of the paper is on simplicial complexes. We demonstrate how to perform signal processing with the framework using numerical examples.
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