Virtual Talk with Michael Schaub

Abstract

In many applications, we are confronted with signals defined on the nodes of a graph.  Think for instance of a sensor network measuring temperature; or a social network, in which each person (node) has an opinion about a specific issue.  Graph signal processing (GSP) tries to device appropriate tools to process such data by generalizing classical methods from signal processing of time-series and images -- such as smoothing, filtering and interpolation -- to signals defined on graphs. Typically, this involves leveraging the structure of the graph as encoded in the spectral properties of the graph Laplacian matrix.

In other applications such as traffic network analysis, however, the signals of interest are naturally defined on the edges of a graph, rather than on the nodes. After a recap of the central ideas of GSP, we examine why the standard tools from GSP may not be suitable for the analysis of such edge signals.  More specifically, we discuss how the underlying notion of a 'smooth signal' inherited from typically considered graph Laplacians are not suitable when dealing with edge signals that encode a notion of flow.  To overcome this limitation we devise signal processing tools based on the Hodge-Laplacian and the associated discrete Hodge Theory for simplicial (and cellular) complexes. We discuss applications of these ideas for signal smoothing, semi-supervised and active learning for edge-flows on discrete (or discretized) spaces.

Bio

Michael Schaub studied Electrical Engineering and Information Technology at ETH Zurich. After an MSc in Biomedical Engineering at Imperial College London, he obtained his PhD in Mathematics at Imperial College London. In the following he worked as a Research Fellow in Belgium, before he moved to the Massachusetts Institute of Technology (MIT) as a Postdoctoral Research Associate. From July 2017 onwards he was a Marie Skłodowska Curie Fellow at MIT and the University of Oxford, before joining RWTH Aachen University in June 2020. He was awarded an ERC Starting grant in 2022.