# Low-complexity Graph Sampling with Noise and Signal Reconstruction via Neumann Series

## TL;DR

This paper introduces a low-complexity graph sampling method using Neumann series and fast Fourier transforms, enabling efficient node selection and robust signal reconstruction in noisy and dynamic networks.

## Contribution

It proposes a novel greedy sampling strategy based on an approximated A-optimality criterion via Neumann series and FGFT, improving efficiency and robustness in graph signal sampling.

## Key findings

- Sampling strategy outperforms existing methods in experiments.
- Reconstruction method is more noise-robust than least squares.
- Efficient node selection without large matrix inversions.

## Abstract

Graph sampling addresses the problem of selecting a node subset in a graph to collect samples, so that a K-bandlimited signal can be reconstructed in high fidelity. Assuming an independent and identically distributed (i.i.d.) noise model, minimizing the expected mean square error (MMSE) leads to the known A-optimality criterion for graph sampling, which is expensive to compute and difficult to optimize. In this paper, we propose an augmented objective based on Neumann series that well approximates the original criterion and is amenable to greedy optimization. Specifically, we show that a shifted A-optimal criterion can be equivalently written as a function of an ideal low-pass (LP) graph filter, which in turn can be approximated efficiently via fast graph Fourier transform (FGFT). Minimizing the new objective, we select nodes greedily without large matrix inversions using a matrix inverse lemma. Further, for the dynamic network case where node availability varies across time, we propose an extended sampling strategy that replaces offline samples one-by-one in the selected set. For signal reconstruction, we propose an accompanied biased signal recovery strategy that reuses the approximated filter from sampling. Experiments show that our reconstruction is more robust to large noise than the least square (LS) solution, and our sampling strategy far outperforms several existing schemes.