Weak detection in the spiked Wigner model

TL;DR

This paper introduces a data-driven hypothesis test for weak detection in a rank-one spiked Wigner matrix, achieving optimality in Gaussian noise and improved performance with non-Gaussian noise through transformations.

Contribution

It develops a new spectral statistic-based test for weak detection that is optimal for Gaussian noise and adaptable to non-Gaussian noise, with theoretical guarantees.

Findings

Test matches likelihood ratio in Gaussian case

Entrywise transformation improves detection in non-Gaussian noise

Central limit theorem established for spectral statistics

Abstract

We consider the weak detection problem in a rank-one spiked Wigner data matrix where the signal-to-noise ratio is small so that reliable detection is impossible. We propose a hypothesis test on the presence of the signal by utilizing the linear spectral statistics of the data matrix. The test is data-driven and does not require prior knowledge about the distribution of the signal or the noise. When the noise is Gaussian, the proposed test is optimal in the sense that its error matches that of the likelihood ratio test, which minimizes the sum of the Type-I and Type-II errors. If the density of the noise is known and non-Gaussian, the error of the test can be lowered by applying an entrywise transformation to the data matrix. We establish a central limit theorem for the linear spectral statistics of general rank-one spiked Wigner matrices as an intermediate step.