Spectral Phase Transition and Optimal PCA in Block-Structured Spiked
  models
Pierre Mergny Justin Ko Florent Krzakala
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.