# Greedy Sampling of Graph Signals

## TL;DR

This paper develops universal performance bounds for noisy graph signal sampling and demonstrates that greedy algorithms can effectively approximate optimal sampling sets despite the problem's combinatorial complexity.

## Contribution

It introduces the concept of approximate submodularity to provide near-optimal guarantees for greedy sampling in graph signal processing.

## Key findings

- Universal bounds for Bayesian estimation from noisy samples.
- Greedy sampling achieves near-optimal performance with explicit bounds.
- Application to reducing complexity in kernel PCA.

## Abstract

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.