Weak detection in the spiked Wigner model

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.

45 pages, 5 figures

References in corpus (7)

• Finite sample approximation results for principal component analysis: a matrix perturbation approach
• Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models
• Fundamental limits of detection in the spiked Wigner model
• Ferromagnetic to paramagnetic transition in spherical spin glass
• Optimal hypothesis testing for stochastic block models with growing degrees
• Weak Detection in the Spiked Wigner Model with General Rank
• On fluctuations of global and mesoscopic linear eigenvalue statistics of generalized Wigner matrices

Cited by in corpus (3)

• Fluctuations Of Linear Spectral Statistics Of Deformed Wigner Matrices
• Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices
• Precise Error Rates for Computationally Efficient Testing