Generalized noise sensitivity of eigenvectors: All eigenvectors, inhomogeneous variance profiles, and dependent resampling

TL;DR

This paper extends the understanding of eigenvector decorrelation under various resampling and noise processes for random matrices, including inhomogeneous and dependent resampling, providing new theoretical insights into eigenvector sensitivity.

Contribution

It generalizes previous results by analyzing eigenvector decorrelation under continuous, discrete, inhomogeneous, and dependent resampling processes, using a novel proof approach.

Findings

Eigenvectors decorrelate after sufficient resampling or time.

Generalized results to inhomogeneous variance profiles.

Analyzed dependent block resampling effects.

Abstract

Chatterjee (2016) proved, as an application of his general framework relating superconcentration and chaos, that after the entries of an $n \times n$ matrix drawn from the Gaussian unitary ensemble undergo an entrywise Ornstein-Uhlenbeck (OU) process for time greater than $n^{-1/3}$, the top eigenvector of the matrix becomes almost completely decorrelated from its initial position. More recently, Bordenave, Lugosi, and Zhivotovskiy (2020) showed that the same happens under a discrete resampling model, once more than $n^{5/3}$ randomly chosen entries of a Wigner random matrix are resampled. We generalize these results in several directions: (1) we analyze the decorrelation of any eigenvector under continuous and discrete resampling dynamics, (2) we analyze the discrete resampling process for generalized Wigner matrices with inhomogeneous variance profiles, (3) we analyze a combination of continuous and discrete resampling where an OU process is repeatedly run for a certain time on randomly chosen entries, and (4) we analyze a dependent version of resampling where entries grouped into "blocks" of arbitrary shapes are resampled together. In each case, we show that a given eigenvector decorrelates provided that enough entries have been resampled or that the associated dynamics have been run for a long enough time. Our proofs take a different approach from prior work, relying more directly on the characterization of eigenvectors as derivatives of eigenvalues and reducing the problem of establishing eigenvector noise sensitivity to variants of standard and robust properties of random matrices such as bounds on eigenvalue spacings and eigenvector delocalization.