Spectral Phase Transition and Optimal PCA in Block-Structured Spiked Models
We discuss the inhomogeneous Wigner spike 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 (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.
2024
Proceedings of Machine Learning Research; 235
2640-3498
35470
35491
Link to the paper
REVIEWED
Event name | Event acronym | Event place | Event date |
ICML | Vienna, Austria | 2024-07-21 - 2024-07-27 | |