Near-optimality of greedy set selection in the sampling of graph signals

Abstract

Sampling has been extensively studied in graph signal processing, having found applications in estimation, clustering, and video compression. Still, sampling set selection remains an open issue. Indeed, although conditions for graph signal reconstruction from noiseless samples were derived, the presence of noise makes sampling set selection combinatorial and NP-hard in general. Performance bounds are available only for randomized sampling schemes, even though greedy search remains ubiquitous in practice. This work sets out to justify the success of greedy sampling by introducing the concept of approximate supermodularity and updating the classical greedy bound to account for this class of functions. Then, it quantifies the approximate supermodularity of two important reconstruction figures of merit, namely the log det of the error covariance matrix and the mean-square error, showing that they can be optimized with worst-case guarantees using greedy sampling.