Sampling of Signals on Multigraphs: From Removable and Uniqueness sets to Spectral Proxies
Abstract: In this paper, we study the sampling of bandlimited signals on multigraphs, where several graphs with different shift operators are defined on the same set of nodes. Since the shift operators do not commute in general, they admit only a joint block diagonalization, and the frequencies of the multigraph signal model are matrices rather than scalars. Relying on a formal algebraic convolutional signal model, we introduce a notion of frequency and bandwidth for multigraph signals that does not depend on any individual layer, and we extend the notions of removable and uniqueness sets to multigraphs. We prove that the optimality of a sampling set is determined exclusively by the common Fourier basis and its invariant subspaces, and not by any individual layer, although the individual layers provide computable sufficient conditions that certify a sampling set as a uniqueness set. Since identifying which layers provide reliable certificates requires spectral information, we propose an aggregate operator that pools the information of all the layers, is computable directly from the shift operators, and certifies uniqueness without any assumption on the individual layers. This operator leads to a greedy sampling algorithm that does not require computing the joint block diagonalization. Numerical experiments on multigraphs whose layers order the frequencies inconsistently show that the proposed algorithm selects better-conditioned sampling sets and attains lower reconstruction errors than single-layer methods.