Signal Recovery on Graphs: Fundamental Limits of Sampling Strategies

Abstract: This paper builds theoretical foundations for the recovery of a newly proposed class of smooth graph signals, approximately bandlimited graph signals, under three sampling strategies: uniform sampling, experimentally designed sampling and active sampling. We then state minimax lower bounds on the maximum risk for the approximately bandlimited class under these three sampling strategies and show that active sampling cannot fundamentally outperform experimentally designed sampling. We propose a recovery strategy to compare uniform sampling with experimentally designed sampling. As the proposed recovery strategy lends itself well to statistical analysis, we derive the exact mean square error for each sampling strategy. To study convergence rates, we introduce two types of graphs and find that (1) the proposed recovery strategy achieves the optimal rates; and (2) the experimentally designed sampling fundamentally outperforms uniform sampling for Type-2 class of graphs. To validate our proposed recovery strategy, we test it on five specific graphs: a ring graph with $k$ nearest neighbors, an Erd\H{o}s-R\'enyi graph, a random geometric graph, a small-world graph and a power-law graph and find that experimental results match the proposed theory well. This work also presents a comprehensive explanation for when and why sampling for semi-supervised learning with graphs works.