Higher-order signal processing with the Dirac operator

2018 52nd Asilomar Conference on Signals, Systems, and Computers, pp. 925–929

Abstract

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.

Authors 3

1. Lucille Calmon

   Queen Mary University of London

   Affiliation as printed

   School of Mathematical Sciences Queen Mary University of London,London,United Kingdom

   School of Mathematical Sciences Queen Mary University of London, London, United Kingdom

2. Michael T. Schaub Aachen Department of Computer Science

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University,Department of Computer Science,Aachen,Germany

   Department of Computer Science, RWTH Aachen University, Aachen, Germany

3. Ginestra Bianconi

   Queen Mary University of London

   Affiliation as printed

   School of Mathematical Sciences Queen Mary University of London,London,United Kingdom

   School of Mathematical Sciences Queen Mary University of London, London, United Kingdom