Sampling of graph signals via randomized local aggregations

Sampling of signals defined over the nodes of a graph is one of the crucial problems in graph signal processing. While in classical signal processing sampling is a well defined operation, when we consider a graph signal many new challenges arise and defining an efficient sampling strategy is not straightforward. Recently, several works have addressed this problem. The most common techniques select a subset of nodes to reconstruct the entire signal. However, such methods often require the knowledge of the signal support and the computation of the sparsity basis before sampling. Instead, in this paper we propose a new approach to this issue. We introduce a novel technique that combines localized sampling with compressed sensing. We first choose a subset of nodes and then, for each node of the subset, we compute random linear combinations of signal coefficients localized at the node itself and its neighborhood. The proposed method provides theoretical guarantees in terms of reconstruction and stability to noise for any graph and any orthonormal basis, even when the support is not known.Comment: IEEE Transactions on Signal and Information Processing over Networks, 201

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

Kernel-based Inference of Functions over Graphs

The study of networks has witnessed an explosive growth over the past decades with several ground-breaking methods introduced. A particularly interesting -- and prevalent in several fields of study -- problem is that of inferring a function defined over the nodes of a network. This work presents a versatile kernel-based framework for tackling this inference problem that naturally subsumes and generalizes the reconstruction approaches put forth recently by the signal processing on graphs community. Both the static and the dynamic settings are considered along with effective modeling approaches for addressing real-world problems. The herein analytical discussion is complemented by a set of numerical examples, which showcase the effectiveness of the presented techniques, as well as their merits related to state-of-the-art methods.Comment: To be published as a chapter in `Adaptive Learning Methods for Nonlinear System Modeling', Elsevier Publishing, Eds. D. Comminiello and J.C. Principe (2018). This chapter surveys recent work on kernel-based inference of functions over graphs including arXiv:1612.03615 and arXiv:1605.07174 and arXiv:1711.0930