Weak Detection in the Spiked Wigner Model with General Rank

Understand

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.