Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies

Abstract

We study the problem of selecting the best sampling set for bandlimited reconstruction of signals on graphs. A frequency domain representation for graph signals can be defined using the eigenvectors and eigenvalues of variation operators that take into account the underlying graph connectivity. Smoothly varying signals defined on the nodes are of particular interest in various applications, and tend to be approximately bandlimited in the frequency basis. Sampling theory for graph signals deals with the problem of choosing the best subset of nodes for reconstructing a bandlimited signal from its samples. Most approaches to this problem require a computation of the frequency basis (i.e., the eigenvectors of the variation operator), followed by a search procedure using the basis elements. This can be impractical, in terms of storage and time complexity, for real datasets involving very large graphs. We circumvent this issue in our formulation by introducing quantities called graph spectral proxies, defined using the powers of the variation operator, in order to approximate the spectral content of graph signals. This allows us to formulate a direct sampling set selection approach that does not require the computation and storage of the basis elements. We show that our approach also provides stable reconstruction when the samples are noisy or when the original signal is only approximately bandlimited. Furthermore, the proposed approach is valid for any choice of the variation operator, thereby covering a wide range of graphs and applications. We demonstrate its effectiveness through various numerical experiments.

Detecting Localized Categorical Attributes on Graphs

“…Finally, we optimize over t by evaluating each pair of t and x * t in the objective function (10). Because of the convex relaxation, the final solution of x is not binary and a higher value of x i indicates a higher confidence that the ith node is activated.…”

“…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.…”

Inference of spatiotemporal processes over graphs via kernel kriged Kalman filtering

“…The tests compare the following reconstruction algorithms: (i) The least mean-squares (LMS) algorithm in [9] with step size µ LMS ; (ii) the distributed least-squares reconstruction (DLSR) algorithm [8] with step sizes µ DLSR and β DLSR (both LMS and DLSR can track slowly time-varying B-bandlimited graph signals); (iii) The B-bandlimited instantaneous estimator (BL-IE) which uses the estimator in [3], [4] per slot t; and (iv) Algorithm 1 with the following configuration: a diffusion kernel (cf. Table I) with parameter σ; a state noise covariance Σ η [t] = s η Σ η with parameter s η > 0 and Σ η := N N a positive definite matrix with N ∈ R N ×N a random matrix with standardized Gaussian entries; and a transition matrix…”

“…The community of signal processing on graphs mainly adopts the so-called bandlimited model, which postulates that the signal of interest lies in a B-dimensional subspace related to the graph topology [3], [4], or assumes that the signal can be sparsely represented on…”

Graph Vertex Sampling with Arbitrary Graph Signal Hilbert Spaces

“…First, in Sec. 2, we show how the sampling approach of [5] can be generalized to any inner product. In particular, our generalization describes a method that can applied to many other GSP tools to extend them to arbitrary Hilbert spaces, not just sampling set selection.…”