BBP transition and the leading eigenvector of the spiked Wigner model with inhomogeneous noise
2026-04-20 • Information Theory
Information Theory
AI summaryⓘ
The authors study a mathematical model used to detect hidden signals in noisy data, called the spiked Wigner model, but with noise that varies unpredictably across the data. They find exact formulas that describe when a hidden signal can be detected based on the noise's properties, identifying a phase transition between detectable and undetectable signals. When the noise has a specific type of variation following a truncated power-law, the authors show that this variability can actually help in detecting the hidden signal better. Their results deepen understanding of how irregular noise impacts signal detection in high-dimensional data.
spiked Wigner modelhigh-dimensional inferencespectral edgeseigenvalue outliersBBP transitionnoise variancerank-one perturbationpower-law distributionsignal detectabilityrandom matrix theory
Authors
Leonardo S. Ferreira, Fernando L. Metz
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.