Spectral Phase Transition and Optimal PCA in Block-Structured Spiked models

TL;DR

This paper extends the BBP phase transition criterion to inhomogeneous, block-structured Wigner models, identifying the optimal spectral method for signal detection and eigenvector recovery in structured noise settings.

Contribution

It provides a rigorous analysis of spectral properties in inhomogeneous Wigner models and establishes the optimality of a spectral method for signal detection.

Findings

Identifies the phase transition threshold for outlier eigenvalues.

Shows the spectral method's optimality in eigenvector recovery.

Extends BBP transition to inhomogeneous, block-structured models.

Abstract

We discuss the inhomogeneous spiked Wigner model, a theoretical framework recently introduced to study structured noise in various learning scenarios, through the prism of random matrix theory, with a specific focus on its spectral properties. Our primary objective is to find an optimal spectral method and to extend the celebrated \cite{BBP} (BBP) phase transition criterion -- well-known in the homogeneous case -- to our inhomogeneous, block-structured, Wigner model. We provide a thorough rigorous analysis of a transformed matrix and show that the transition for the appearance of 1) an outlier outside the bulk of the limiting spectral distribution and 2) a positive overlap between the associated eigenvector and the signal, occurs precisely at the optimal threshold, making the proposed spectral method optimal within the class of iterative methods for the inhomogeneous Wigner problem.