Subgraph Signal Processing

Graph signal processing, like the graph Fourier transform, requires the fullgraph signal at every vertex of the graph. However, in practice, only signalsat a subset of vertices may be available. We propose a subgraph signalprocessing framework that relates a graph shift operator or filter on asubgraph with a filter on the ambient graph through an operator loss. It allowsus to define shift operators for the subgraph signal, which has a meaningfulinterpretation and relation to mixtures of shift invariant filters restrictedto different subgraphs of the ambient graph (which we call semi shiftinvariant). This leads to a notion of frequency domain for the subgraph signalconsistent in some sense with that of the full graph signal. We apply thesubgraph signal processing machinery to several applications and demonstratethe utility of this framework in cases where only partial graph signals areobserved.