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

Xiangyi Zhu, Dmitriy Kunisky

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.

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

Abstract

Chatterjee (2016) proved, as an application of his general framework relating superconcentration and chaos, that after the entries of an matrix drawn from the Gaussian unitary ensemble undergo an entrywise Ornstein-Uhlenbeck (OU) process for time greater than , 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 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.
Paper Structure (42 sections, 44 theorems, 290 equations)

This paper contains 42 sections, 44 theorems, 290 equations.

Table of Contents

  1. Introduction
  2. Overview of main results
  3. Overview of proof techniques
  4. Main results
  5. Ornstein-Uhlenbeck process on GOE matrices
  6. Poisson-driven block Ornstein-Uhlenbeck process on GOE matrices
  7. Block resampling on generalized Wigner matrices
  8. Proof techniques
  9. Comparison with previous techniques
  10. Organization
  11. Preliminaries
  12. Notation
  13. Markov processes and semigroups
  14. Variance identities
  15. Ornstein-Uhlenbeck process
  16. ...and 27 more sections

Key Result

Theorem 1.7

Let $G(t) \sim \mathrm{OU}(n, 1)$ and $\alpha = \alpha(n) \in [n]$. Suppose that $t = t(n)$ is such that Then, we have

Theorems & Definitions (92)

  • Definition 1.2: Gaussian orthogonal ensemble
  • Definition 1.3: Sub-Gaussian generalized Wigner matrix
  • Remark 1.4: Justification of assumptions
  • Definition 1.5: Dyson Brownian motion
  • Definition 1.6: Matrix Ornstein-Uhlenbeck process
  • Theorem 1.7
  • Definition 1.8: Block size
  • Definition 1.9: Admissible partition
  • Remark 1.10
  • Definition 1.11: Poisson-driven block Ornstein-Uhlenbeck process
  • ...and 82 more