Greedy Sampling of Graph Signals
This work addresses a fundamental challenge in graph signal processing for applications like estimation and compression, offering theoretical guarantees for widely used greedy sampling methods, though it is incremental in improving existing bounds.
The paper tackles the problem of selecting sampling sets for graph signals in the presence of noise, which is a hard combinatorial problem with no existing performance guarantees for greedy methods. It derives universal performance bounds for Bayesian estimation from noisy samples and provides near-optimal guarantees for greedy sampling by introducing approximate submodularity, showing that greedy search can optimize mean-square error with worst-case guarantees.
Sampling is a fundamental topic in graph signal processing, having found applications in estimation, clustering, and video compression. In contrast to traditional signal processing, the irregularity of the signal domain makes selecting a sampling set non-trivial and hard to analyze. Indeed, though conditions for graph signal interpolation from noiseless samples exist, they do not lead to a unique sampling set. The presence of noise makes choosing among these sampling sets a hard combinatorial problem. Although greedy sampling schemes are commonly used in practice, they have no performance guarantee. This work takes a twofold approach to address this issue. First, universal performance bounds are derived for the Bayesian estimation of graph signals from noisy samples. In contrast to currently available bounds, they are not restricted to specific sampling schemes and hold for any sampling sets. Second, this paper provides near-optimal guarantees for greedy sampling by introducing the concept of approximate submodularity and updating the classical greedy bound. It then provides explicit bounds on the approximate supermodularity of the interpolation mean-square error showing that it can be optimized with worst-case guarantees using greedy search even though it is not supermodular. Simulations illustrate the derived bound for different graph models and show an application of graph signal sampling to reduce the complexity of kernel principal component analysis.