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Comparison theorems for the extreme eigenvalues of a random symmetric matrix

Joel A. Tropp

Abstract

This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian random matrix that inherits its statistics from the sum, and it strengthens previous results of this type. Corollaries address the minimum eigenvalue and the spectral norm. The comparison methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper improves on existing eigenvalue bounds for random matrices arising in spectral graph theory, quantum information theory, high-dimensional statistics, and numerical linear algebra. In particular, these techniques deliver the first complete proof that a sparse random dimension reduction map has the injectivity properties conjectured by Nelson & Nguyen in 2013.

Comparison theorems for the extreme eigenvalues of a random symmetric matrix

Abstract

This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian random matrix that inherits its statistics from the sum, and it strengthens previous results of this type. Corollaries address the minimum eigenvalue and the spectral norm. The comparison methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper improves on existing eigenvalue bounds for random matrices arising in spectral graph theory, quantum information theory, high-dimensional statistics, and numerical linear algebra. In particular, these techniques deliver the first complete proof that a sparse random dimension reduction map has the injectivity properties conjectured by Nelson & Nguyen in 2013.
Paper Structure (58 sections, 16 theorems, 188 equations)

This paper contains 58 sections, 16 theorems, 188 equations.

Table of Contents

  1. Main results
  2. The independent sum model
  3. The Gaussian proxy model
  4. Matrix fluctuation statistics
  5. Comparison: Maximum eigenvalue of a self-adjoint matrix
  6. Remarks
  7. Example: Wigner's matrix
  8. Unbounded summands
  9. Comparison: Minimum eigenvalue of a self-adjoint matrix
  10. Example: Rademacher covariance matrix
  11. Comparison: Spectral norm of a rectangular matrix
  12. Example: Spectral norm of a Rademacher matrix
  13. Related work
  14. Matrix concentration
  15. Universality for independent sums
  16. ...and 43 more sections

Key Result

Theorem 1.1

Consider an independent family $(\bm{W}_1, \dots, \bm{W}_n)$ of random self-adjoint matrices with common dimension $d$ and with two finite moments. Assume that the random matrices satisfy the uniform upper bound Form the sum and its Gaussian proxy: Then the maximum eigenvalues admit the comparison Furthermore, for a parameter $s \geq \log d$, the tail probability satisfies the bound The statis

Theorems & Definitions (30)

  • Theorem 1.1: Comparison: Maximum eigenvalue of an independent sum
  • Corollary 1.2: Comparison: Maximum eigenvalue, unbounded summands
  • Corollary 1.3: Comparison: Minimum eigenvalue of an independent sum
  • proof
  • Corollary 1.4: Comparison: Spectral norm of an independent sum
  • proof
  • Remark 1.5: Universality
  • Theorem 3.1: Random regular graph: Second eigenvalue
  • Theorem 3.2: Random Pauli model
  • Theorem 3.3: Sample covariance: Fourth-moment theorem
  • ...and 20 more