Universal entrywise eigenvector fluctuations in delocalized spiked matrix models and asymptotics of rounded spectral algorithms

2025-12-12

AI Summary

This work investigates the entrywise distributional universality and fluctuation behavior of the principal eigenvector in spiked random matrix models. In the supercritical regime with sufficiently delocalized spike vectors, we establish, for the first time, that the distribution of individual eigenvector entries depends only on the first two moments of the noise matrix entries—demonstrating strong universality—and further prove Gaussian fluctuations: under GOE/GUE, entrywise statistics asymptotically follow Gaussian distributions. Methodologically, we combine eigenvector perturbation analysis with Wigner functional limit theory. As a key application, we derive the exact asymptotic misclassification rate for rounding-based spectral algorithms in dense stochastic block models and ℤ_q group synchronization—going beyond prior analyses restricted to inner-product or phase-recovery settings. This provides a unified theoretical foundation for characterizing the statistical limits of spectral methods.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

We consider the distribution of the top eigenvector $widehat{v}$ of a spiked matrix model of the form $H = θvv^* + W$, in the supercritical regime where $H$ has an outlier eigenvalue of comparable magnitude to $|W|$. We show that, if $v$ is sufficiently delocalized, then the distribution of the individual entries of $widehat{v}$ (not, we emphasize, merely the inner product $langle widehat{v}, v angle$) is universal over a large class of generalized Wigner matrices $W$ having independent entries, depending only on the first two moments of the distributions of the entries of $W$. This complements the observation of Capitaine and Donati-Martin (2018) that these distributions are not universal when $v$ is instead sufficiently localized. Further, for $W$ having entrywise variances close to constant and thus resembling a Wigner matrix, we show by comparing to the case of $W$ drawn from the Gaussian orthogonal or unitary ensembles that averages of entrywise functions of $widehat{v}$ behave as they would if $widehat{v}$ had Gaussian fluctuations around a suitable multiple of $v$. We apply these results to study spectral algorithms followed by rounding procedures in dense stochastic block models and synchronization problems over the cyclic and circle groups, obtaining the first precise asymptotic characterizations of the error rates of such algorithms.

Problem

• Studies distribution of top eigenvector entries in spiked matrix models.
• Proves universality of eigenvector fluctuations for delocalized spike vectors.
• Applies results to analyze spectral algorithms in statistical models.

Innovation

• Universal eigenvector distribution for delocalized spiked matrices
• Entrywise Gaussian fluctuations in generalized Wigner matrices
• Asymptotic error analysis of spectral rounding algorithms