Sampling and Reconstruction of Graph Signals via Weak Submodularity and Semidefinite Relaxation

TL;DR

This paper addresses the challenge of optimally sampling bandlimited graph signals in noisy environments by formulating the problem as a weak submodular maximization and semidefinite relaxation, providing efficient algorithms with performance guarantees.

Contribution

It introduces a novel approach combining weak submodularity and SDP relaxation for graph signal sampling, along with a randomized greedy algorithm with proven performance bounds.

Findings

The randomized greedy algorithm significantly speeds up sampling with minimal MSE loss.

The proposed methods outperform existing greedy schemes in speed while maintaining comparable accuracy.

Numerical simulations validate the effectiveness of the approach on synthetic and real-world graphs.

Abstract

We study the problem of sampling a bandlimited graph signal in the presence of noise, where the objective is to select a node subset of prescribed cardinality that minimizes the signal reconstruction mean squared error (MSE). To that end, we formulate the task at hand as the minimization of MSE subject to binary constraints, and approximate the resulting NP-hard problem via semidefinite programming (SDP) relaxation. Moreover, we provide an alternative formulation based on maximizing a monotone weak submodular function and propose a randomized-greedy algorithm to find a sub-optimal subset. We then derive a worst-case performance guarantee on the MSE returned by the randomized greedy algorithm for general non-stationary graph signals. The efficacy of the proposed methods is illustrated through numerical simulations on synthetic and real-world graphs. Notably, the randomized greedy algorithm yields an order-of-magnitude speedup over state-of-the-art greedy sampling schemes, while incurring only a marginal MSE performance loss.