Detection of signal in the rank-k spiked Wigner model

Abstract

We study the statistical decision process about detecting the presence of signal from a 'signal+noise' type matrix model with an additive Wigner noise. If the signal-to-noise ratio (SNR) is greater than the suitable value (called as a reconstruction threshold), then we can implement a reliable detection (known as a strong detection.) However, below that value, a strong type detection is impossible. In the latter case, we propose a specific hypothesis test that utilizes the linear spectral statistics of the data matrix (known as a weak detection.) Moreover, we obtain an error of the likelihood ratio test under the Gaussian noise, which is an optimal error by the Neymann-Pearson lemma, and it matches the error of the proposed test i.e., our test is optimal. Additionally, we can observe that our test does not depend on the distribution of the signal or the noise. The first part of our proof is based on the Gaussian interpolation method and the cavity method, as devised by Guerra and Talagrand in their study of the Sherrington--Kirkpatrick (SK) spin glass model. The second part is based on a central limit theorem for the linear spectral statistics of the general rank-$k$ spiked Wigner matrices.