Signal Recovery on Graphs: Fundamental Limits of Sampling Strategies

Abstract

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 Erdős-Rényi 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.

Optimal sampling strategies for adaptive learning of graph signals

“…Finally, we compare the sampling strategy in (14) with some established sampling methods for graph signals, namely, the leverage score sampling from [15], the Max-Det greedy strategy from [16], and the (uniformly) random sampling. For each strategy, we keep adding nodes to the sampling set according to the corresponding criterion until the constraints on the learning rate and the MSD in (14) are satisfied.…”

“…A very hot topic in GSP is the development of a sampling theory for graph signals, which was initially considered in [6], and later extended in [9], [8], [10], [11], [12]. Then, several reconstruction methods have been proposed, either iterative as in [13], [14], or batch, as in [8], [10], [15]. Recently, adaptive strategies for online reconstruction and learning of graph signals were also proposed in [16]- [18], and paved the way to the development of novel adaptive GSP tools.…”

Detecting Localized Categorical Attributes on Graphs

“…3 The graph scan statistic searches over graphs and localizes the localized attribute and is a data-dependent and generative approach, which works for both detection and localization.…”

“…Massive amounts of data being generated from various sources including social networks, citation, biological, and physical infrastructure have spurred the emerging area of analyzing data supported on graphs [1], [2] giving rise to a variety of scientific and engineering studies; for example, selecting representative training data to improve semi-supervised learning with graphs [3]; detecting communities in communication or social networks [4]; ranking the most important websites on the Internet [5]; and detecting anomalies in sensor networks [6].…”

Deep Unsupervised Learning of 3D Point Clouds via Graph Topology Inference and Filtering

“…The framework models underlying structure by a graph and signals by graph signals, generalizing concepts and tools from classical discrete signal processing to the graph domain. Some techniques involve representations for graph signals [24], [25], sampling for graph signals [26], [27], [28], recovery for graph signals [29], [30], denoising [31], [32], graph-based filter banks [31], [33], graphbased transforms [34], [24], graph topology inference [35], and graph neural networks [36], [37]. To process 3D point clouds, [38] uses graph filters and graph-based resampling strategies to select most informative 3D nodes; [39] uses graph convolutional neural networks to classify 3D point clouds.…”