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Efficient error and variance estimation for randomized matrix computations

Ethan N. Epperly, Joel A. Tropp

TL;DR

This document serves as a comprehensive guide to the SIAM LaTeX style, detailing available class options, front matter conventions, cross-referencing and hyperlinking features, and built-in support for math, theorems, figures, tables, and algorithms. It also covers managing supplementary materials, template files, and bibliographic enhancements, including DOIs, URLs, arXiv identifiers, and software citations. The guide emphasizes automation for consistent SIAM formatting, accessibility through PDF bookmarks, and extensibility via special macros and appendix handling. Overall, it enables authors to prepare SIAM-compliant manuscripts with robust referencing, templating, and publication-ready features.

Abstract

Randomized matrix algorithms have become workhorse tools in scientific computing and machine learning. To use these algorithms safely in applications, they should be coupled with posterior error estimates to assess the quality of the output. To meet this need, this paper proposes two diagnostics: a leave-one-out error estimator for randomized low-rank approximations and a jackknife resampling method to estimate the variance of the output of a randomized matrix computation. Both of these diagnostics are rapid to compute for randomized low-rank approximation algorithms such as the randomized SVD and randomized Nyström approximation, and they provide useful information that can be used to assess the quality of the computed output and guide algorithmic parameter choices.

Efficient error and variance estimation for randomized matrix computations

TL;DR

This document serves as a comprehensive guide to the SIAM LaTeX style, detailing available class options, front matter conventions, cross-referencing and hyperlinking features, and built-in support for math, theorems, figures, tables, and algorithms. It also covers managing supplementary materials, template files, and bibliographic enhancements, including DOIs, URLs, arXiv identifiers, and software citations. The guide emphasizes automation for consistent SIAM formatting, accessibility through PDF bookmarks, and extensibility via special macros and appendix handling. Overall, it enables authors to prepare SIAM-compliant manuscripts with robust referencing, templating, and publication-ready features.

Abstract

Randomized matrix algorithms have become workhorse tools in scientific computing and machine learning. To use these algorithms safely in applications, they should be coupled with posterior error estimates to assess the quality of the output. To meet this need, this paper proposes two diagnostics: a leave-one-out error estimator for randomized low-rank approximations and a jackknife resampling method to estimate the variance of the output of a randomized matrix computation. Both of these diagnostics are rapid to compute for randomized low-rank approximation algorithms such as the randomized SVD and randomized Nyström approximation, and they provide useful information that can be used to assess the quality of the computed output and guide algorithmic parameter choices.
Paper Structure (29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem