Fast MSE-Based Sampling of Bandlimited Graph Signals via Low-Pass Impulse Responses
Fen Wang, Gene Cheung, Minxiang Ye, Taihao Li, Y Feng · IEEE Transactions on Signal Processing · 2023
Sampling is a fundamental problem in graph signal processing that selects a node subset to collect samples, so that data in the remaining nodes can be well recovered. Existing eigen-decomposition-free (ED-free) graph sampling schemes are not designed to minimize mean square error (MMSE) of reconstructed bandlimited graph signals, often resulting in sub-par MSE performance. In this paper, we propose a lightweight ED-free algorithm to minimize an approximate MSE objective using per-node impulse responses of an ideal low-pass graph filter. Specifically, we first derive a proxy of the MMSE greedy sampling objective without matrix inverse. We then define node-dependent impulse responses of an ideal low-pass graph filter—analogous to the sinc function in traditional signal processing—which can be approximated very fast without ED. We use these response vectors to reformulate our derived sampling objective, then show that the optimal sampling set has supported low-pass impulse responses that are as orthogonal as possible—defaulting to uniform sampling for 1D regular kernels when the graph Fourier basis is DFT. For optimization, we propose a fast sampling algorithm to evaluate candidates via vector-vector multiplications by reusing previous greedy results. For faster sampling, we relax the MMSE-based greedy objective to a bounded approximation, so that candidate nodes can be easily appraised using simple scalar multiplications. Extensive experiments show that our sampling method achieved state-of-the-art sampling speed and had the best MSE performance among deterministic ED-free sampling methods in various scenarios.