Greedy Algorithm with Approximation Ratio for Sampling Noisy Graph Signals

Abstract

We study the optimal sampling set selection problem in sampling a noisy $k$ -bandlimited graph signal. To minimize the effect of noise when trying to reconstruct a $k$ -bandlimited graph signal from $m$ samples, the optimal sampling set selection problem has been shown to be equivalent to finding a $m\times k$ submatrix with the maximum smallest singular value, $\sigma_{\min}$ [3]. As the problem is NP-hard, we present a greedy algorithm inspired by a similar submatrix selection problem known in computer science and to which we add a local search refinement. We show that 1) in experiments, our algorithm finds a submatrix with larger $\sigma_{\min}$ than prior greedy algorithm [3], and 2) has a proven worst-case approximation ratio of $1/(1+\epsilon)k$ , where $\epsilon$ is a constant.