Fast Sampling of Graph Signals with Noise via Neumann Series Conversion

Graph sampling with independent noise towards minimum mean square error (MMSE) leads to the known A-optimality criterion, which is computation-intensive to evaluate and NP-hard to optimize. In this paper, we propose a new low-complexity sampling strategy based on Neumann series that circumvents large matrix inversion and eigen-decomposition. We first prove that a DC-shifted A-optimality criterion is equivalent to an objective computed using the inverse of a sub-matrix of an ideal graph low-pass (LP) filter. The LP filter matrix can be approximated efficiently via fast Graph Fourier Transform (FGFT). Using the shifted A-optimality objective as a proxy, we then propose a fast algorithm to greedily select samples one-by-one based on a matrix inversion lemma with simple matrix updates. We show that the obtained solution has a performance upper bound via super-modularity analysis. Simulation results show that our proposed sampling strategy has lower complexity and outperforms several existing deterministic sampling schemes.