Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Matrix perturbation bounds via contour bootstrapping

Phuc Tran, Van Vu

TL;DR

This paper uses a "contour bootstrapping" argument to derive several new perturbation bounds on the error occurring when one uses matrix sparsification to speed up the computation of spectral parameters.

Abstract

Matrix perturbation bounds play an essential role in the design and analysis of spectral algorithms. In this paper, we use a "contour bootstrapping" argument to derive several new perturbation bounds. As applications, we discuss new bounds on the error occurring when one uses matrix sparsification to speed up the computation of spectral parameters. Another potential application is the estimation of the trade-off in computing with privacy.

Matrix perturbation bounds via contour bootstrapping

TL;DR

This paper uses a "contour bootstrapping" argument to derive several new perturbation bounds on the error occurring when one uses matrix sparsification to speed up the computation of spectral parameters.

Abstract

Matrix perturbation bounds play an essential role in the design and analysis of spectral algorithms. In this paper, we use a "contour bootstrapping" argument to derive several new perturbation bounds. As applications, we discuss new bounds on the error occurring when one uses matrix sparsification to speed up the computation of spectral parameters. Another potential application is the estimation of the trade-off in computing with privacy.
Paper Structure (24 sections, 22 theorems, 118 equations)

This paper contains 24 sections, 22 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Perturbation Theory
  3. Eigenspace Perturbation
  4. Low rank approximation
  5. The contour bootstrapping argument
  6. New Perturbation Bounds
  7. Eigenspace Perturbation
  8. Low rank approximation
  9. General $f$
  10. Applications
  11. Stable rank of a matrix
  12. Fast computation with random sparsification
  13. Computing the leading eigenvector
  14. Computing low rank approximation
  15. Proofs of the main theorems
  16. ...and 9 more sections

Key Result

Theorem 1.2.1

We have

Theorems & Definitions (27)

  • Remark 1.1.1
  • Theorem 1.2.1: Davis-Kahan DKoriginalBook1
  • Corollary 1.2.2
  • Remark 1.2.3
  • Theorem 1.3.1: Eckart-Young
  • Theorem 1.3.2: Weyl We1Book1
  • Remark 1.3.3
  • Remark 1.3.4
  • Lemma 2.0.1
  • Theorem 3.1.1: Eigenspace Perturbation
  • ...and 17 more