Towards Accelerated Greedy Sampling and Reconstruction of Bandlimited Graph Signals

TL;DR

This paper introduces efficient greedy algorithms for sampling and reconstructing bandlimited graph signals, significantly reducing computational costs while maintaining high accuracy in large-scale settings.

Contribution

It proposes accelerated greedy sampling methods for large-scale graph signals, with theoretical guarantees for noiseless and noisy cases, and formulates the problem as a submodular optimization.

Findings

Exact recovery in noiseless case with iterative sampling

Bounded reconstruction error in noisy measurements

Performance guarantees for the greedy algorithm

Abstract

We study the problem of sampling and reconstructing spectrally sparse graph signals where the objective is to select a subset of nodes of prespecified cardinality that ensures interpolation of the original signal with the lowest possible reconstruction error. This task is of critical importance in Graph signal processing (GSP) and while existing methods generally provide satisfactory performance, they typically entail a prohibitive computational cost when it comes to the study of large-scale problems. Thus, there is a need for accelerated and efficient methods tailored for high-dimensional and large-scale sampling and reconstruction tasks. To this end, we first consider a non-Bayesian scenario and propose an efficient iterative node sampling procedure that in the noiseless case enables exact recovery of the original signal from the set of selected nodes. In the case of noisy measurements, a bound on the reconstruction error of the proposed algorithm is established. Then, we consider the Bayesian scenario where we formulate the sampling task as the problem of maximizing a monotone weak submodular function, and propose a randomized-greedy algorithm to find a sub-optimal subset of informative nodes. We derive worst-case performance guarantees on the mean-square error achieved by the randomized-greedy algorithm for general non-stationary graph signals.