Spectral Phase Transition and Optimal PCA in Block-Structured Spiked models
This work addresses theoretical challenges in structured noise models for machine learning, providing rigorous analysis for optimal spectral methods, though it is incremental as it builds on existing spiked model frameworks.
The authors tackled the problem of optimal spectral methods in block-structured spiked Wigner models with inhomogeneous noise, extending the BBP phase transition criterion to this setting and proving that the transition occurs at the optimal threshold, making their method optimal within iterative methods for this problem.
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.