Multiplier Bootstrap and Edge Phase Transitions of High-Dimensional Covariance Matrices

What the math gives to ML

The paper identifies a nonstandard failure mode of multiplier bootstrap for extreme eigenvalues in high-dimensional covariance problems: the bootstrap law can switch between Tracy–Widom, Gaussian, Weibull, Fréchet, and Gumbel regimes depending on multiplier tails, aspect ratio, and population spikes. The transferable asset is a phase-aware spectral diagnostic that separates genuine low-rank spikes from bulk fluctuations in finite-width neural representations. A practical neural-network use is to apply bounded multiplier resampling to activation or gradient covariance matrices, estimate whether leading eigenvalues are statistically exceptional, and activate low-rank preconditioning or representation regularization only when supported by the diagnostic. The paper’s resolvent sign-change construction also suggests a computational outlier detector based on a scalar secular equation rather than repeatedly computing every eigenvalue.