🤖 AI Summary
Classical spectral perturbation theory, such as the Davis–Kahan theorem, fails to capture the systematic geometric bias in the leading eigenspace under heterogeneous noise. This work studies a signal-plus-noise model with heteroscedastic and sparse random perturbations, and for the first time identifies and quantifies a deterministic geometric bias induced by the alignment between the signal structure and the noise variance profile. The total perturbation is decomposed into three components: a signal-to-noise ratio term, a stochastic fluctuation term, and this geometric bias term. Leveraging the quadratic vector equation (QVE) framework and refined isotropic local laws, the paper establishes non-asymptotic perturbation bounds in both operator norm and ℓ²→ℓ^∞ norm. The resulting upper bounds are nearly optimal and substantially outperform classical theory in predicting eigenspace perturbations under heterogeneous noise.
📝 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.