QR factorization-based sampling set selection for bandlimited graph signals

Abstract

We consider the problem of sampling vertices or nodes of graphs where signals on graphs are bandlimited and reconstructed from the signal values on the sampled nodes. Noting that sampling a subset of nodes is equivalent to constructing the sampled matrix consisting of the corresponding rows selected from the eigenvector matrix of a variation operator (e.g., graph Laplacian), we first formulate the reconstruction error as the Frobenius matrix norm of the pseudo-inverse of the sampled matrix and manipulate the reconstruction error by using QR factorization to propose a simple criterion by which greedy sampling is conducted. We show that the proposed algorithm can achieve near-optimality for the case of high SNR with the concept of approximate supermodularity and evaluate the proposed method by comparing with a near-optimal approach developed for a sensor selection problem. We also analyze the complexity of the proposed algorithm in operation count and compare with existing greedy methods, including algorithms for subset selection of matrices since sampling of graph signals is also accomplished by selecting a subset of columns from the transpose of the eigenvector matrix. We finally demonstrate through extensive experiments that the proposed algorithm achieves a competitive performance gain for signals on various graphs in terms of complexity and signal fidelity as compared with previous novel methods.