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Extremal bounds for Gaussian trace estimation

Eric Hallman

Abstract

This work derives extremal tail bounds for the Gaussian trace estimator applied to a real symmetric matrix. We define a partial ordering on the eigenvalues, so that when a matrix has greater spectrum under this ordering, its estimator will have worse tail bounds. This is done for two families of matrices: positive semidefinite matrices with bounded effective rank, and indefinite matrices with bounded 2-norm and fixed Frobenius norm. In each case, the tail region is defined rigorously and is constant for a given family.

Extremal bounds for Gaussian trace estimation

Abstract

This work derives extremal tail bounds for the Gaussian trace estimator applied to a real symmetric matrix. We define a partial ordering on the eigenvalues, so that when a matrix has greater spectrum under this ordering, its estimator will have worse tail bounds. This is done for two families of matrices: positive semidefinite matrices with bounded effective rank, and indefinite matrices with bounded 2-norm and fixed Frobenius norm. In each case, the tail region is defined rigorously and is constant for a given family.

Paper Structure

This paper contains 24 sections, 11 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Extremal bounds
  3. The tail
  4. The extremal matrices
  5. Extension to Gamma random variables
  6. Related work
  7. Majorization
  8. Indefinite majorization
  9. Gamma random variables
  10. Basic properties
  11. Generalizing the distribution families
  12. Infinite division
  13. Fixed shape and scale parameters
  14. Majorization theorems
  15. Relative error
  16. ...and 9 more sections

Key Result

Theorem 1

Let ${\bf A}\in \mathbb{R}^{n\times n}$ be symmetric. Then for all $\varepsilon > 0$,

Theorems & Definitions (32)

  • Theorem 1: cortinovis2021indefinite, Thm. 1
  • Definition 1
  • Corollary 1: cortinovis2021indefinite, Remark 2
  • Definition 2
  • Definition 3
  • Lemma 1
  • Definition 4
  • Definition 5
  • Lemma 2
  • Definition 6
  • ...and 22 more