Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Superfast 1-Norm Estimation

Soo Go, Victor Y. Pan

TL;DR

Novel superfast matrix algorithms are devised that consistently produce accurate 1-norm estimates for real-world matrices and some promising extensions of these surprisingly simple techniques can potentially be adopted in practical computations.

Abstract

A matrix algorithm is said to be superfast (that is, runs at sublinear cost) if it involves much fewer scalars and flops than the input matrix has entries. Such algorithms have been extensively studied and widely applied in modern computations for matrices with low displacement rank and more recently for low-rank approximation of matrices, even though they are known to fail on worst-case inputs in the latter application. We devise novel superfast algorithms that consistently produce accurate 1-norm estimates for real-world matrices and discuss some promising extensions of our surprisingly simple techniques. With further testing and refinement, our algorithms can potentially be adopted in practical computations.

Superfast 1-Norm Estimation

TL;DR

Novel superfast matrix algorithms are devised that consistently produce accurate 1-norm estimates for real-world matrices and some promising extensions of these surprisingly simple techniques can potentially be adopted in practical computations.

Abstract

A matrix algorithm is said to be superfast (that is, runs at sublinear cost) if it involves much fewer scalars and flops than the input matrix has entries. Such algorithms have been extensively studied and widely applied in modern computations for matrices with low displacement rank and more recently for low-rank approximation of matrices, even though they are known to fail on worst-case inputs in the latter application. We devise novel superfast algorithms that consistently produce accurate 1-norm estimates for real-world matrices and discuss some promising extensions of our surprisingly simple techniques. With further testing and refinement, our algorithms can potentially be adopted in practical computations.

Paper Structure

This paper contains 35 sections, 16 equations, 9 figures, 5 tables, 6 algorithms.

Table of Contents

  1. Key Words:
  2. 2020 Math. Subject Classification:
  3. Introduction
  4. Superfast matrix computations: the State of the Art (briefly)
  5. Our results
  6. Some extensions
  7. Related work
  8. Organization of the paper
  9. Background for 1-norm estimation
  10. 1-norm estimators with randomized sparsification
  11. Outline
  12. A sparsified 1-Norm Estimator without scaling
  13. Computational cost estimates
  14. Sparsified 1-norm estimation with scaling
  15. 1-norm estimation incorporating C-A iterations
  16. ...and 20 more sections

Figures (9)

  • Figure 1: The first three steps of maxvol 1 output three entries marked with strips.
  • Figure 2: LRA of $M$.
  • Figure 3: Output errors of Alg. \ref{['algnormest2']}.
  • Figure 4: Number of iterations of Alg. \ref{['algnormest2']}.
  • Figure 5: Output errors of Alg. \ref{['algnormest']}.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1
  • Example A.1
  • Remark A.1
  • Remark C.1
  • Remark C.2