Sampling of Graph Signals With Successive Local Aggregations

Abstract

Abstract-A new scheme to sample signals defined in 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. Most of the works that have looked at this problem have focused on using the value of the signal observed at a subset of nodes to recover the signal in the entire graph. Differently, 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, which is critical to guarantee recovery 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. The last part of the paper presents numerical experiments that illustrate the results developed through both synthetic graph signals and a real-world graph of the economy of the United States.

Space-shift sampling of graph signals

“…The aim is to have at least K linearly independent equations, so that the matrix C(I ⊗ (ΨEK ))Ῡ has full column rank and the original signal x can be recovered using the pseudoinverse. Beyond invertibility, taking additional samples improves the recovery performance in the presence of noise [9]. Although space limitations prevent us to present the details here, the structure in (5) can be used to design optimal sampling and recovery schemes that minimize the effects of the noise.…”

“…Nevertheless, identifiability is guaranteed whenever C(I ⊗ Ψ)Υ is full spark and contains at least 2K rows [15]. For specific forms of C, the full-spark condition can be assessed by a quick inspection of {λi} N i=1 and V; see [9,10] for the case of aggregation sampling.…”

“…Most existing works have focused on observing the value of the signal at a subset of nodes to recover the signal in the entire graph [5][6][7][8]. By contrast, a scheme where the samples are taken at a single node that aggregates local information sequentially was recently proposed in [9][10][11]. Here, we present a more general sampling scheme that contains both mentioned paradigms as particular cases.…”

Detecting Localized Categorical Attributes on Graphs

“…It extends classical signal processing concepts such as signals, filters, Fourier transform, frequency response, low-and highpass filtering, from signals residing on regular lattices to data residing on general graphs; for example, a graph signal models the data value assigned to each node in a graph. Recent work involves sampling for graph signals [9], [10], [11], [12], recovery for graph signals [13], [14], [15], [16], representations for graph signals [17], [18] principles on graphs [19], [20], stationary graph signal processing [21], [22], graph dictionary construction [23], graph-based filter banks [24], [25], [26], [27], denoising on graphs [24], [28], community detection and clustering on graphs [29], [30], [31], distributed computing [32], [33] and graph-based transforms [34], [35], [36]. We here consider detecting localized categorical attributes on graphs.…”

Reconstruction of graph signals: Percolation from a single seeding node

“…Inspired by the local aggregation sampling procedure in [5], we propose two reconstruction -interpolation -schemes where the signal is injected at a single node and percolates throughout the graph via successive applications of the graph shift operator. In Sec.…”