Tail Bounds for All Eigenvalues of a Sum of Random Matrices

Abstract

The field of nonasymptotic random matrix theory has traditionally focused on the problem of bounding the extreme eigenvalues of a random matrix. In some circumstances, however, we may also be interested in studying the behavior of the interior eigenvalues. In this case, classical tools do not readily apply. Indeed, the interior eigenvalues are determined by the minmax of a random process, which is very challenging to control. This paper demonstrates that it is possible to combine the matrix Laplace transform method detailed in [Tro11c] with the Courant{Fischer characterization of eigenvalues to obtain nontrivial bounds on the interior eigenvalues of a sum of random self-adjoint matrices. This approach expands the scope of the matrix probability inequalities from [Tro11c] so that they provide interesting information about the bulk spectrum.

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Document Details

Document Type
Technical Report
Publication Date
Jul 21, 2011
Accession Number
ADA563094

Entities

People

  • Alex Gittens
  • Joel A. Tropp

Organizations

  • California Institute of Technology

Tags

Communities of Interest

  • Energy and Power Technologies
  • Materials and Manufacturing Processes

DTIC Thesaurus Topics

  • Accuracy
  • Banach Space
  • Convergence
  • Covariance
  • Data Science
  • Eigenvalues
  • Functional Analysis
  • Information Science
  • Matrix Theory
  • Probability
  • Random Variables
  • Signal Processing
  • Spectra
  • Standards
  • Statistics
  • Stochastic Processes
  • Theoretical Computer Science

Readers

  • Linear Algebra
  • Mathematical Modeling and Probability Theory.
  • Systems Analysis and Design