BBP transition and the leading eigenvector of the spiked Wigner model with inhomogeneous noise
Provides theoretical insights into high-dimensional inference under inhomogeneous noise, relevant for statistical physicists and machine learning theorists.
The authors study an inhomogeneous spiked Wigner model with random noise variances, deriving exact equations for spectral edges, outlier eigenvalue, and eigenvector components. They find that inhomogeneous noise can enhance signal detectability, as shown by a non-monotonic BBP transition line.
The spiked Wigner ensemble is a prototypical model for high-dimensional inference. We study the spectral properties of an inhomogeneous rank-one spiked Wigner model in which the variance of each entry of the noise matrix is itself a random variable. In the high-dimensional limit, we derive exact equations for the spectral edges, the outlier eigenvalue, and the distribution of the components of the outlier eigenvector. These equations determine the BBP transition line that separates the gapped phase, where the signal is detectable, from the gapless phase. In the gapped regime, the distribution of the outlier eigenvector provides a natural estimator of the spike. We solve the equations for a noise matrix whose variances are generated from a truncated power-law distribution. In this case, the BBP transition line is non-monotonic, showing that an inhomogeneous noise can enhance signal detectability.