Michael Schaub Seminar at the Department of Engineering Science, University of Oxford

Abstract: We are confronted with signals defined on the nodes of a graph in many applications.  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.

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 very brief 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 (the typically considered variants of) the graph Laplacian are not suitable when dealing with edge signals that encode flows.  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.

Short Bio: I studied Electrical Engineering and Information Technology at ETH Zurich with a focus on communication systems. After a MSc in Biomedical Engineering at Imperial College London (Neurotechnology stream), I moved to the Mathematics Department to obtain my PhD under the supervision of Prof. Mauricio Barahona and Prof. Sophia Yaliraki. Following my PhD, I worked in Belgium, jointly at the Université catholique de Louvain and at the University of Namur, as a Postdoctoral Research Fellow. In November 2016, I moved to the Massachusetts Institute of Technology (MIT) as a Postdoctoral Research Associate. From July 2017 onwards I was a Marie Skłodowska Curie Fellow at MIT and the University of Oxford, before joining RWTH Aachen University in June 2020, supported by the NRW Return Programme (2019).