Geometric bias in eigenspace perturbation under random heterogeneous noise

• URL: http://arxiv.org/abs/2606.11263v1
• Date: Tue, 09 Jun 2026 03:50:14 GMT
• Title: Geometric bias in eigenspace perturbation under random heterogeneous noise
• Abstract summary: Spectral methods rely on the stability of principal eigenspaces under random perturbations.<n>Classically, this stability is quantified by the Davis-Kahan and Wedin theorems.<n>We show that under heterogeneous noise variances, the empirical eigenvectors suffer a systematic, deterministic geometric bias.
• Score: 2.5484436411433573
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Spectral methods rely fundamentally on the stability of principal eigenspaces under random perturbations. Classically, this stability is quantified by the Davis-Kahan and Wedin theorems, which bound the eigenspace error using the operator norm of the noise and the relevant spectral gaps. While these worst-case bounds are sharp for arbitrary deterministic perturbations, they can be wasteful in the low-rank signal-plus-random-noise setting, as they fail to capture the fine-grained interaction between the signal geometry and the noise distribution. In this paper, we study the spectral perturbation of signal-plus-noise matrices corrupted by sparse, random noise with an arbitrary, inhomogeneous variance profile. We demonstrate that under heterogeneous noise variances, the empirical eigenvectors suffer a systematic, deterministic geometric bias that is entirely invisible to classical perturbation bounds. By leveraging the Quadratic Vector Equation (QVE) and establishing fine-grained isotropic local laws, we derive near-optimal, non-asymptotic perturbation bounds for the leading eigenspaces in the operator and $2\to\infty$ norms. The bounds 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.

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