Sampling of signals defined over the nodes of a graph is one of the crucial problems in graph signal processing, whereas 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.
@cite_8 , motivated by an analogy with classical sampling, propose a sampling strategy computing measurements as the result of iterated applications of the graph shift operator (e.g., the adjacency matrix) observed at one or more nodes. This strategy can guarantee perfect reconstruction with the optimal number of measurements and does not require knowledge of the support of the signal, so it can also be used when the support is unknown. The technique is equivalent to linear sampling where the sampling matrix has a Vandermonde structure. Unfortunately, Vandermonde matrices are known to be poorly conditioned @cite_21 resulting in severe instability even for moderately-sized graphs. The analogy drawn by the authors with classical sampling does not consider that the Vandermonde structure arising in classical sampling is a unique special case that is well conditioned, as it results in the DFT @cite_21 . Moreover, the method is highly dependent on the specific nodes chosen as observation points and it is not clear how to perform the selection.
@cite_21 @cite_8
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Estimation of the condition numbers of Vandermonde matrices, motivated by applications to interpolation and quadrature, can be traced back to at least the 1970s. Empirical study has consistently shown that Vandermonde matrices tend to be badly ill-conditioned, with a narrow class of exceptions, such as the matrices of the discrete Fourier transform (hereafter we use the acronym DFT). So far, however, formal support for this empirical observation has been limited to the matrices defined by the real set of knots. We prove that, more generally, any Vandermonde matrix of a large size is badly ill-conditioned unless its knots are more or less equally spaced on or about the circle @math . The matrices of the DFT are unitary up to scaling and therefore perfectly conditioned. They are defined by a cyclic sequence of knots, equally spaced on that circle, but we prove that even a slight modification of the knots into the so-called quasi-cyclic sequence on this circle defines badly ill-conditioned Vandermond... A new scheme to sample signals defined on the nodes of a graph is proposed. The underlying assumption is that such signals admit a sparse representation in a frequency domain related to the structure of the graph, which is captured by the so-called graph-shift operator. Instead of using the value of the signal observed at a subset of nodes to recover the signal in the entire graph, the sampling scheme proposed here uses as input observations taken at a single node. The observations correspond to sequential applications of the graph-shift operator, which are linear combinations of the information gathered by the neighbors of the node. When the graph corresponds to a directed cycle (which is the support of time-varying signals), our method is equivalent to the classical sampling in the time domain. When the graph is more general, we show that the Vandermonde structure of the sampling matrix, critical when sampling time-varying signals, is preserved. Sampling and interpolation are analyzed first in the absence of noise, and then noise is considered. We then study the recovery of the sampled signal when the specific set of frequencies that is active is not known. Moreover, we present a more general sampling scheme, under which, either our aggregation approach or the alternative approach of sampling a graph signal by observing the value of the signal at a subset of nodes can be both viewed as particular cases. Numerical experiments illustrating the results in both synthetic and real-world graphs close the paper.