论文标题
与狄拉克运算符的高阶信号处理
Higher-order signal processing with the Dirac operator
论文作者
论文摘要
最近出现了由节点，边缘和高阶细胞定义的简单和细胞复合物上的信号的处理，最近出现了作为对更通用拓扑空间支持的信号的图形信号处理的原则扩展。但是，到目前为止，大多数作品都考虑了仅考虑适当定义的移位操作员（例如图形laplacian或hodge laplacian）来考虑与单个类型单元相关的信号（例如节点信号或边缘信号的处理）的信号处理问题。在这里，我们介绍了狄拉克操作员，作为一种新型的转移操作员，用于复合物上的信号处理。我们讨论了狄拉克操作员如何具有密切的关系，但与霍奇 - 拉普拉斯人不同并检查其光谱特性。重要的是，狄拉克操作员以原则上的方式在相邻维度上定义的信号。我们演示了这是如何使我们能够利用节点信号来处理边缘流量的。
The processing of signals on simplicial and cellular complexes defined by nodes, edges, and higher-order cells has recently emerged as a principled extension of graph signal processing for signals supported on more general topological spaces. However, most works so far have considered signal processing problems for signals associated to only a single type of cell such as the processing of node signals, or edge signals, by considering an appropriately defined shift operator, like the graph Laplacian or the Hodge Laplacian. Here we introduce the Dirac operator as a novel kind of shift operator for signal processing on complexes. We discuss how the Dirac operator has close relations but is distinct from the Hodge-Laplacian and examine its spectral properties. Importantly, the Dirac operator couples signals defined on cells of neighboring dimensions in a principled fashion. We demonstrate how this enables us, e.g., to leverage node signals for the processing of edge flows.
