Identifying First-Order Lowpass Graph Signals Using Perron Frobenius Theorem

Abstract

This paper is concerned with the blind identification of graph filters from graph signals. Our aim is to determine if the graph filter generating the graph signals is first-order lowpass without knowing the graph topology. Notice that lowpass graph filter is a common prerequisite for applying graph signal processing tools for sampling, denoising, and graph learning. Our method is inspired by the Perron Frobenius theorem, which observes that for first-order lowpass graph filter, the top eigenvector of output covariance would be the only eigenvector with elements of the same sign. Utilizing this observation, we develop a simple detector that answers if a given data set is produced by a first-order lowpass graph filter. We analyze the effects of finite-sample, graph size, observation noise, strength of lowpass filter, on the detector's performance. Numerical experiments on synthetic and real data support our findings.

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“…We show how to set the threshold of the proposed LRT and semi-parametric detectors by providing analytical/approximated terms for the associated probability of false alarms. The proposed methods require fewer data than the first-order graph LPF method [38] and do not assume an ideal graph LPF as the BSMSD [40]. Simulation results verify that the proposed detectors detect smooth graph signals and demonstrate that the proposed methods outperform existing methods on the tested scenarios in terms of detection performance.…”

“…However, usually, these works do not validate the smoothness assumption. In [38], the authors suggest a first-order low-pass graph signal detector for data sets whose social/physical models are unknown. However, the detector cannot be modified to higher-order low-pass graph signals.…”

“…The solution in (38) holds under the assumption that all the eigenvalues of L are distinct. Otherwise, the graph filter should satisfy the constraint in (5).…”

“…Otherwise, the graph filter should satisfy the constraint in (5). As a result, the ML estimator in (38) should be replaced by the constrained ML estimator. In Appendix E it is shown that the constrained ML is given by…”

“…We compare the detection performance of the proposed methods with the performance of 1) The first-order LPF detector [38], which assumes a measurement model similar to (17) and aims to determine if the graph filter is a first-order low-pass graph signal, without using the graph topology. Thus, it requires an extensive dataset to estimate the data covariance matrix.…”