Fluctuations of the largest eigenvalues of transformed spiked Wigner matrices

We consider a spiked random matrix model obtained by applying a function entrywise to a signal-plus-noise symmetric data matrix. We prove that the largest eigenvalue of this model, which we call a transformed spiked Wigner matrix, exhibits Baik-Ben Arous-Péché (BBP) type phase transition. We show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy-Widom distribution when the effective SNR is below the critical number. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes.

When a smal piece of useful information is hidden within a lot of random noise, it can be hard to distinguish the signal from the pure noise. The principal component analysis (PCA), a well-established tool in statistics based on a mathematical object known as the largest eigenvalue, is one of the most important algorithm to detect the presence of the signal from the noisy data. In this work, we explore quantatively how the features of the PCA should change if we know how the noisy data has been transformed. Our results can provide theoretical background for the understanding of noisy data naturally appearing in various fields of study, including the theoretical analysis of machine learning.