Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies

We study the problem of selecting the best sampling set for bandlimited reconstruction of signals on graphs. A frequency domain representation for graph signals can be defined using the eigenvectors and eigenvalues of variation operators that take into account the underlying graph connectivity. Smoothly varying signals defined on the nodes are of particular interest in various applications, and tend to be approximately bandlimited in the frequency basis. Sampling theory for graph signals deals with the problem of choosing the best subset of nodes for reconstructing a bandlimited signal from its samples. Most approaches to this problem require a computation of the frequency basis (i.e., the eigenvectors of the variation operator), followed by a search procedure using the basis elements. This can be impractical, in terms of storage and time complexity, for real datasets involving very large graphs. We circumvent this issue in our formulation by introducing quantities called graph spectral proxies, defined using the powers of the variation operator, in order to approximate the spectral content of graph signals. This allows us to formulate a direct sampling set selection approach that does not require the computation and storage of the basis elements. We show that our approach also provides stable reconstruction when the samples are noisy or when the original signal is only approximately bandlimited. Furthermore, the proposed approach is valid for any choice of the variation operator, thereby covering a wide range of graphs and applications. We demonstrate its effectiveness through various numerical experiments.