BBP transition and the leading eigenvector of the spiked Wigner model with inhomogeneous noise

TL;DR

This paper analyzes the spectral properties of an inhomogeneous spiked Wigner model, deriving exact equations for phase transitions and eigenvector distributions, revealing how inhomogeneous noise can improve signal detection.

Contribution

It provides the first detailed analysis of the BBP transition in inhomogeneous noise settings, including explicit solutions for power-law distributed variances.

Findings

The BBP transition line can be non-monotonic with inhomogeneous noise.

In the gapped phase, the outlier eigenvector effectively estimates the spike.

Inhomogeneous noise can enhance the detectability of signals.

Abstract

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.