Subspace Change Point Detection Under Spiked Wigner Model

Abstract

Detecting the emergence of a spike signal from the Wigner noise is a fundamental problem of interest in many application areas, such as covert communication, inference of genetic population structure, and network community detection. We consider this problem within the framework of non-Bayesian change-point detection. We study the Gaussian case first. A scan-statistic akin to the Shewhart chart is proposed, based on the operator norm of the sample matrix. By assuming the observation dimension grows to infinity, we could characterize the average run length (ARL) and the probability of missing detection (Type-II error) through the asymptotic theory of random matrices; under the non-asymptotic circumstances, we establish a sharp lower bound on the ARL. In a contrast, we derive the exact subspace-CUSUM procedure with complete information about the model parameter and introduce the subspace-CUSUME algorithm to handle the situation with some unidentified parameter. We recapitulate some optimal properties relative to the classical CUSUM statistic. The spectral-based algorithm could be extended to a class of non-Gaussian models where entries of the Wigner noise follow i.i.d sub-exponential distributions. We present part of comparable analysis results as in the Gaussian case and discuss the qualitative relationships among ARL and other parameters in depth. Finally, we corroborate corresponding theoretical conclusions through numerical simulations.