Discrete Signal Processing on Graphs: Sampling Theory

TL;DR

This paper develops a sampling theory for signals on graphs, enabling perfect recovery of bandlimited signals and introducing methods for robust sampling, with applications to semi-supervised classification tasks.

Contribution

It introduces a graph sampling theory that generalizes classical sampling, including optimal sampling operators and a graph filter bank, for improved signal recovery and classification.

Findings

Perfect recovery for bandlimited graph signals.

Random sampling achieves high-probability recovery on certain graph types.

Application to semi-supervised classification with fewer labels.

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.