Weak detection in the spiked Wigner model
This work addresses a fundamental statistical inference problem for researchers in signal processing and random matrix theory, offering an incremental improvement in detection methods.
The paper tackles the weak detection problem in rank-one spiked Wigner matrices with low signal-to-noise ratios, proposing a data-driven hypothesis test using linear spectral statistics that is optimal for Gaussian noise and improvable for known non-Gaussian noise.
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.