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Norms of Randomized Circulant Matrices

Rafał Latała, Witold Świątkowski

Abstract

We investigate two-sided bounds for operator norms of random matrices with unhomogenous independent entries. We formulate a lower bound for Rademacher matrices and conjecture that it may be reversed up to a universal constant. We show that our conjecture holds up to $\log\log n$ factor for randomized $n\times n$ circulant matrices and double logarithm may be eliminated under some mild additional assumptions on the coefficients.

Norms of Randomized Circulant Matrices

Abstract

We investigate two-sided bounds for operator norms of random matrices with unhomogenous independent entries. We formulate a lower bound for Rademacher matrices and conjecture that it may be reversed up to a universal constant. We show that our conjecture holds up to factor for randomized circulant matrices and double logarithm may be eliminated under some mild additional assumptions on the coefficients.

Paper Structure

This paper contains 6 sections, 18 theorems, 144 equations.

Table of Contents

  1. Introduction and main results
  2. Proofs of main results
  3. Proof of Theorem \ref{['thm:lower']}
  4. Proof of Theorem \ref{['thm:twosidedbound']}
  5. Estimates for Rademacher norms
  6. Extensions

Key Result

Theorem \oldthetheorem

Let $(a_{ij})_{i,j\le n}$ be any real matrix and $X_{ij}=a_{ij}\varepsilon_{ij}$. Then

Theorems & Definitions (40)

  • Theorem \oldthetheorem
  • Conjecture \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • proof : Proof of Theorem \ref{['thm:lower']}
  • Lemma \oldthetheorem
  • proof
  • ...and 30 more