Geometric bias in eigenspace perturbation under random heterogeneous noise

Abstract

Spectral methods rely on the stability of principal eigenspaces under random perturbations. Classically, this is quantified by the Davis-Kahan and Wedin theorems, which bound the eigenspace error via the operator norm of the noise and the relevant spectral gaps. While sharp for arbitrary deterministic perturbations, these worst-case bounds can be wasteful in the low-rank signal-plus-noise setting, as they fail to capture the interaction between the signal geometry and the noise distribution. We study the spectral perturbation of signal-plus-noise matrices corrupted by sparse random noise with an arbitrary, inhomogeneous variance profile. Under heterogeneous variances, the empirical eigenvectors suffer a systematic, deterministic geometric bias invisible to classical bounds. Leveraging the Quadratic Vector Equation (QVE) and fine-grained isotropic local laws, we derive near-optimal, non-asymptotic bounds for the leading eigenspaces in the operator and 2-to-infinity norms. These separate the usual signal-to-noise contribution, stochastic fluctuations, and structured geometric bias terms determined by the alignment between the signal eigenspaces and the row-wise variance profile. We further develop refined rowwise bounds that adapt to the variance-weighted leverage of the signal space, yielding sharper guarantees in delocalized regimes. As applications, we establish strong consistency of adjacency spectral clustering for degree-corrected stochastic block models with heterogeneous degrees and unbalanced communities, recovering the logarithmic expected-degree scale in the regular balanced case. We also study spectral embedding for generalized random dot product graphs, showing that the full signal embedding admits sharp rowwise control, whereas spectral truncation can retain a systematic geometric bias determined by the omitted signal directions and the variance profile.

132 pages, 2 figures. Restructured the paper and added new applications of the main perturbation results