Random sampling of bandlimited signals on graphs

We study the problem of sampling k-bandlimited signals on graphs. We propose two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure, and its performance depends on a parameter called the graph coherence. On the contrary, the second strategy is adaptive but yields optimal results. Indeed, no more than O(k log(k)) measurements are sufficient to ensure an accurate and stable recovery of all k-bandlimited signals. This second strategy is based on a careful choice of the sampling distribution, which can be estimated quickly. Then, we propose a computationally efficient decoder to reconstruct k-bandlimited signals from their samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we conduct several experiments to test these techniques

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

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