Randomized extended Kaczmarz for solving least squares

Anastasios Zouzias, Nikolaos Freris

Research output: Contribution to journalArticle

Abstract

We present a randomized iterative algorithm that exponentially converges in the mean square to the minimum ℓ2-norm least squares solution of a given linear system of equations. The expected number of arithmetic operations required to obtain an estimate of given accuracy is proportional to the squared condition number of the system multiplied by the number of nonzero entries of the input matrix. The proposed algorithm is an extension of the randomized Kaczmarz method that was analyzed by Strohmer and Vershynin.

Original languageEnglish (US)
Pages (from-to)773-793
Number of pages21
JournalSIAM Journal on Matrix Analysis and Applications
Volume34
Issue number2
DOIs
StatePublished - Jul 29 2013

Fingerprint

Least Squares
Least-squares Solution
Linear system of equations
Randomized Algorithms
Condition number
Mean Square
Iterative Algorithm
Directly proportional
Converge
Norm
Estimate

Keywords

  • Iterative method
  • LAPACK
  • Linear least squares
  • Minimum-length solution
  • Overdetermined system
  • Random sampling
  • Randomized algorithms
  • Sparse matrix
  • Underdetermined system

ASJC Scopus subject areas

  • Analysis

Cite this

Randomized extended Kaczmarz for solving least squares. / Zouzias, Anastasios; Freris, Nikolaos.

In: SIAM Journal on Matrix Analysis and Applications, Vol. 34, No. 2, 29.07.2013, p. 773-793.

Research output: Contribution to journalArticle

Zouzias, Anastasios ; Freris, Nikolaos. / Randomized extended Kaczmarz for solving least squares. In: SIAM Journal on Matrix Analysis and Applications. 2013 ; Vol. 34, No. 2. pp. 773-793.
@article{a14d27c410fd45a9afb3193d0d6496fa,
title = "Randomized extended Kaczmarz for solving least squares",
abstract = "We present a randomized iterative algorithm that exponentially converges in the mean square to the minimum ℓ2-norm least squares solution of a given linear system of equations. The expected number of arithmetic operations required to obtain an estimate of given accuracy is proportional to the squared condition number of the system multiplied by the number of nonzero entries of the input matrix. The proposed algorithm is an extension of the randomized Kaczmarz method that was analyzed by Strohmer and Vershynin.",
keywords = "Iterative method, LAPACK, Linear least squares, Minimum-length solution, Overdetermined system, Random sampling, Randomized algorithms, Sparse matrix, Underdetermined system",
author = "Anastasios Zouzias and Nikolaos Freris",
year = "2013",
month = "7",
day = "29",
doi = "10.1137/120889897",
language = "English (US)",
volume = "34",
pages = "773--793",
journal = "SIAM Journal on Matrix Analysis and Applications",
issn = "0895-4798",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "2",

}

TY - JOUR

T1 - Randomized extended Kaczmarz for solving least squares

AU - Zouzias, Anastasios

AU - Freris, Nikolaos

PY - 2013/7/29

Y1 - 2013/7/29

N2 - We present a randomized iterative algorithm that exponentially converges in the mean square to the minimum ℓ2-norm least squares solution of a given linear system of equations. The expected number of arithmetic operations required to obtain an estimate of given accuracy is proportional to the squared condition number of the system multiplied by the number of nonzero entries of the input matrix. The proposed algorithm is an extension of the randomized Kaczmarz method that was analyzed by Strohmer and Vershynin.

AB - We present a randomized iterative algorithm that exponentially converges in the mean square to the minimum ℓ2-norm least squares solution of a given linear system of equations. The expected number of arithmetic operations required to obtain an estimate of given accuracy is proportional to the squared condition number of the system multiplied by the number of nonzero entries of the input matrix. The proposed algorithm is an extension of the randomized Kaczmarz method that was analyzed by Strohmer and Vershynin.

KW - Iterative method

KW - LAPACK

KW - Linear least squares

KW - Minimum-length solution

KW - Overdetermined system

KW - Random sampling

KW - Randomized algorithms

KW - Sparse matrix

KW - Underdetermined system

UR - http://www.scopus.com/inward/record.url?scp=84880534082&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880534082&partnerID=8YFLogxK

U2 - 10.1137/120889897

DO - 10.1137/120889897

M3 - Article

AN - SCOPUS:84880534082

VL - 34

SP - 773

EP - 793

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 2

ER -