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Eigenvalue Stability and New Perturbation Bounds for the extremal eigenvalues of a matrix

Phuc Tran, Van Vu

Abstract

Let $A$ be a full ranked $ n\times n$ matrix, with singular values $σ_1 (A) \ge \dots \ge σ_n (A) >0$. The condition number $κ(A):= σ_1(A)/σ_n(A)=\|A\|\cdot \|A\|^{-1}$ is a key parameter in the analysis of algorithms taking $A$ as input. In practice, matrices (representing real data) are often perturbed by noise. Technically speaking, the real input would be a noisy variant $\tilde A =A +E$ of $A$, where $E$ represents the noise. The condition number $κ(\tilde A)$ will be used instead of $κ(A)$. Thus, it is of importance to measure the impact of noise on the condition number. In this paper, we focus on the case when the noise is random. We introduce the notion of regional stability, via which we design a new framework to estimate the perturbation of the extremal singular values and the condition number of a matrix. Our framework allows us to bound the perturbation of singular values through the perturbation of singular spaces. We then bound the latter using a novel contour analysis argument, which, as a co-product, provides an improved version of the classical Davis-Kahan theorem in many settings. Our new estimates concerning the least singular value $σ_n(A)$ complement well-known results in this area, and are more favorable in the case when the ground matrix $A$ is large compared to the noise matrix $E$.

Eigenvalue Stability and New Perturbation Bounds for the extremal eigenvalues of a matrix

Abstract

Let be a full ranked matrix, with singular values . The condition number is a key parameter in the analysis of algorithms taking as input. In practice, matrices (representing real data) are often perturbed by noise. Technically speaking, the real input would be a noisy variant of , where represents the noise. The condition number will be used instead of . Thus, it is of importance to measure the impact of noise on the condition number. In this paper, we focus on the case when the noise is random. We introduce the notion of regional stability, via which we design a new framework to estimate the perturbation of the extremal singular values and the condition number of a matrix. Our framework allows us to bound the perturbation of singular values through the perturbation of singular spaces. We then bound the latter using a novel contour analysis argument, which, as a co-product, provides an improved version of the classical Davis-Kahan theorem in many settings. Our new estimates concerning the least singular value complement well-known results in this area, and are more favorable in the case when the ground matrix is large compared to the noise matrix .
Paper Structure (25 sections, 23 theorems, 114 equations)

This paper contains 25 sections, 23 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. New results
  3. Bounding eigenvalues via regional stability
  4. Stability of a region.
  5. Regional stability via perturbation of projections
  6. A new stability theorem
  7. Bounding the least singular value via stability: the hole argument
  8. Moving the hole around: perturbation of arbitrary eigenvalues
  9. Bounding leading eigenvalues via stability
  10. Proof of Theorem \ref{['theo: main0']}: the main ideas
  11. Proof of Theorem \ref{['theo: main0']} via Theorem \ref{['theorem: main']}.
  12. Proof of Theorem \ref{['theorem: main']}: Bounding the eigenspace perturbation via Contour analysis
  13. A new way to bound the contour integral: the double jump strategy
  14. Proof of Inequality \ref{['bounding E(z-A)E(z-A)']}
  15. Bounding $\|M_2\|, \|M_3\|, \|M_4\|$.
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose that the entries of $E$ are iid copies of a random variable with mean $0$ and variance $1$, then for any constants $C_1, C_2 > 0$ and $C_3 = 2(C_2 +2) C_1 +1/2 +o(1)$, the following holds. If $\| A \| \le n^{C_1}$, then

Theorems & Definitions (30)

  • Remark 1.0.1
  • Theorem 1.1
  • Remark 2.0.1
  • Theorem 2.1
  • Definition 2.1
  • Lemma 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Remark 2.0.2
  • Corollary 2.4
  • ...and 20 more