Discrete Signal Processing on Graphs: Sampling Theory

Abstract

We propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that perfect recovery is possible for graph signals bandlimited under the graph Fourier transform. The sampled signal coefficients form a new graph signal, whose corresponding graph structure preserves the first-order difference of the original graph signal. For general graphs, an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise; for graphs whose graph Fourier transforms are frames with maximal robustness to erasures as well as for Erd\H{o}s-R\'enyi graphs, random sampling leads to perfect recovery with high probability. We further establish the connection to the sampling theory of finite discrete-time signal processing and previous work on signal recovery on graphs. To handle full-band graph signals, we propose a graph filter bank based on sampling theory on graphs. Finally, we apply the proposed sampling theory to semi-supervised classification on online blogs and digit images, where we achieve similar or better performance with fewer labeled samples compared to previous work.Comment: To appear in IEEE T-S

Space-shift sampling of graph signals

“…Under the selection sampling approach [5][6][7][8], sampling a graph signal amounts to settingx = Cx. Since the K × N binary matrix C selects the observed nodes, the issue then is how to design C, i.e., which nodes to select, and how to recover the original signal x from its samplesx.…”

“…Classical time domain signals can be represented as graph signals defined on a directed cycle [2,7]; see Fig. 1.…”

“…As a general rule, when sampling an approximately bandlimited signal whose active frequencies are associated with large eigenvalues of S -which is a standard assumption in the literature [5][6][7][8] -, AS is expected to give rise to a better recovery. Successive applications of S amplify the active frequencies, entailing a better estimation of these frequencies and reducing the reconstruction error.…”

“…The underlying assumption is that such signals admit a sparse representation in a frequency domain related to the structure of this graph. 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].…”

Interpolation of graph signals using shift-invariant graph filters

“…, P , then it is not possible to reconstruct a signal whose kth frequency coefficient [ x] k is non-zero using that set of nodes. This problem is also present when sampling graph signals by observing the value of the signal in a subset of nodes [9].…”