Joint online estimation of multi-order graph topological signals

In this paper, we propose a general method to process time-varying signals on different orders of simplicial complexes in an online fashion. The proposed Hodge normalized least mean square algorithm (Hodge-NLMS) utilizes spatial and spectral techniques of topological signal processing defined using the Hodge Laplacians to form an online algorithm for signals on either the nodes or the edges of a graph. The joint estimation of a graph with signals coexisting on nodes and edges is also realized through an alternating execution of the Hodge-NLMS on the nodes and edges. Experiment results have confirmed that our proposed methods could accurately track both time-varying node and edge signals on synthetic data generated on top of graphs collected in the real world.