Weak Detection in the Spiked Wigner Model with General Rank
This addresses a statistical detection problem in signal processing or data analysis, but it appears incremental as it builds on existing spiked matrix models with general rank extensions.
The paper tackles the problem of detecting a signal in a spiked Wigner matrix model with general rank, proposing a hypothesis test based on linear spectral statistics that is optimal for Gaussian noise at low signal-to-noise ratios and can be improved for non-Gaussian noise via entrywise transformation, with an algorithm for rank estimation when unknown.
We study the statistical decision process of detecting the signal from a `signal+noise' type matrix model with an additive Wigner noise. We propose a hypothesis test based on the linear spectral statistics of the data matrix, which does not depend on the distribution of the signal or the noise. The test is optimal under the Gaussian noise if the signal-to-noise ratio is small, as it minimizes the sum of the Type-I and Type-II errors. Under the non-Gaussian noise, the test can be improved with an entrywise transformation to the data matrix. We also introduce an algorithm that estimates the rank of the signal when it is not known a priori.