Sampling and Recovery of Graph Signals

Abstract

The aim of this chapter is to give an overview of the recent advances related to sampling and recovery of signals defined over graphs. First, we illustrate the conditions for perfect recovery of bandlimited graph signals from samples collected over a selected set of vertexes. Then, we describe some sampling design criteria proposed in the literature to mitigate the effect of noise and model mismatching when performing graph signal recovery. Finally, we illustrate algorithms and optimal sampling strategies for adaptive recovery and tracking of dynamic graph signals, where both sampling set and signal values are allowed to vary with time. Numerical simulations carried out over both synthetic and real data illustrate the potential advantages of graph signal processing methods for sampling, interpolation, and tracking of signals observed over irregular domains such as, e.g., technological or biological networks. Notation and Background 3maximum graph cuts [21] and spanning trees [22]. Finally, there exist randomized sampling strategies, e.g., [23,24,25]. The work in [23] provides an efficient design of sampling probability distribution over the nodes, deriving bounds on the reconstruction error in the presence of noise and/or approximatively bandlimited signals. Reference [24] exploits compressive sampling arguments to derive random sampling strategies with variable density, thus also proposing a fast technique to estimate the optimal sampling distribution accurately. Last, the work in [25] proposes a sampling strategy tailored for large-scale data based on random walks on graphs.The sampling strategies described so far involve batch methods for sampling and recovery of graph signals. In many applications such as, e.g., transportation networks, brain networks, or communication networks, the observed graph signals are typically time-varying. This requires the development of effective methods capable to learn and track dynamic graph signals from a carefully designed, possibly time-varying, sampling set. Some previous works have considered this specific learning task, see, e.g., [26,27,28,29]. Specifically,[26] proposed an LMS estimation strategy enabling adaptive learning and tracking from a limited number of smartly sampled observations. The LMS method in [26] was then extended to the distributed setting in [27]. The work in [28] proposed a kernel-based reconstruction framework to accommodate time-evolving signals over possibly time-evolving topologies, leveraging spatio-temporal dynamics of the observed data. Finally, reference [29] proposes a distributed method for tracking bandlimited graph signals, assuming perfect observations and a fixed sampling strategy.In this chapter, we review some of the recent advances related to sampling and recovery of signals defined over graphs. Due to space limitations, such review will be limited only to some specific contributions. The structure of the chapter is explained in the sequel. Sec. 1.2 defines the adopted notation, and recalls some background on GSP. In Sec. 1.3, we illu...

Scalable $M$-Channel Critically Sampled Filter Banks for Graph Signals

“…3) A scalable method to sample and interpolate bandpass and highpass graph signals. While we leverage the recent flurry of work in sampling and reconstruction of graph signals [27]- [39], the prior literature focuses on methods that require a full eigendecomposition or assume the graph signals are smooth (lowpass). We use efficient convex optimization methods with a novel penalty term to perform the interpolation (Section III).…”

Joint Forecasting and Interpolation of Time-Varying Graph Signals Using Deep Learning

“…where SV max (.) stands for the largest singular value [77] and S = V \ S. This means that no F-bandlimited signal over the graph G is supported on S.…”

“…In practice, most GSs are only approximately bandlimited [78]. A GS is approximately (F, )-bandlimited if [77] x…”

Graph Topology Inference Based on Sparsifying Transform Learning

“…Then we compare what we are able to reconstruct with what is indeed known. We use a greedy sampling strategy using an E-optimal design criterion [35], selecting the bandwidth provided by our transform learning method, which is equal to 60 for the pre-ictal interval, and to 64 for the ictal interval. The number of electrodes assumed to be observed is chosen equal to the bandwidth in both cases.…”

“…For each time instant, we use a batch consistent recovery method that reconstructs the signals from the collected samples [see eq. (1.9) in [35]]. In Fig.…”