research-article

Fast matrix rank algorithms and applications

Authors:

Lap Chi LauAuthors Info & Claims

Journal of the ACM (JACM), Volume 60, Issue 5

Article No.: 31, Pages 1 - 25

https://doi.org/10.1145/2528404

Published: 28 October 2013 Publication History

Abstract

We consider the problem of computing the rank of an m × n matrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns in Õ(|A| + r^ω) field operations, where |A| denotes the number of nonzero entries in A and ω < 2.38 is the matrix multiplication exponent. Previously the best known algorithm to find a set of r linearly independent columns is by Gaussian elimination, with deterministic running time O(mnr^ω-2). Our algorithm is faster when r < max{m,n}, for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank in Õ(mn) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in exact linear algebra, combinatorial optimization and dynamic data structure.

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Cited By

Cohen K(2025)Powers of low rank sparse matricesTheoretical Computer Science10.1016/j.tcs.2025.1150821032(115082)Online publication date: Mar-2025
https://doi.org/10.1016/j.tcs.2025.115082
Seliverstov AZverkov O(2024)Lower bounds for the rank of a matrix with zeros and ones outside the leading diagonalПрограммирование10.31857/S0132347424020133(100-107)Online publication date: 15-Apr-2024
https://doi.org/10.31857/S0132347424020133
Seliverstov AZverkov O(2024)Lower Bounds for the Rank of a Matrix with Zeros and Ones outside the Leading DiagonalProgramming and Computing Software10.1134/S036176882402014250:2(202-207)Online publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1134/S0361768824020142
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Index Terms

Fast matrix rank algorithms and applications
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations in finite fields

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cover image Journal of the ACM

Journal of the ACM Volume 60, Issue 5

October 2013

245 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2528384

Issue’s Table of Contents

Copyright © 2013 ACM.

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Publication History

Published: 28 October 2013

Accepted: 01 March 2013

Revised: 01 February 2013

Received: 01 June 2012

Published in JACM Volume 60, Issue 5

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Cited By

Cohen K(2025)Powers of low rank sparse matricesTheoretical Computer Science10.1016/j.tcs.2025.1150821032(115082)Online publication date: Mar-2025
https://doi.org/10.1016/j.tcs.2025.115082
Seliverstov AZverkov O(2024)Lower bounds for the rank of a matrix with zeros and ones outside the leading diagonalПрограммирование10.31857/S0132347424020133(100-107)Online publication date: 15-Apr-2024
https://doi.org/10.31857/S0132347424020133
Seliverstov AZverkov O(2024)Lower Bounds for the Rank of a Matrix with Zeros and Ones outside the Leading DiagonalProgramming and Computing Software10.1134/S036176882402014250:2(202-207)Online publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1134/S0361768824020142
Lin JHuang YLee MChen P(2024)Coded Distributed Multiplication for Matrices of Different Sparsity LevelsIEEE Transactions on Communications10.1109/TCOMM.2023.332619372:2(633-647)Online publication date: Feb-2024
https://doi.org/10.1109/TCOMM.2023.3326193
Meier MNakatsukasa Y(2024)Fast randomized numerical rank estimation for numerically low-rank matricesLinear Algebra and its Applications10.1016/j.laa.2024.01.001686(1-32)Online publication date: Apr-2024
https://doi.org/10.1016/j.laa.2024.01.001
Grasegger GLegerský J(2024)Flexibility and rigidity of frameworks consisting of triangles and parallelogramsComputational Geometry: Theory and Applications10.1016/j.comgeo.2023.102055120:COnline publication date: 1-Jun-2024
https://dl.acm.org/doi/10.1016/j.comgeo.2023.102055
Dhar ANatarajan VRathod A(2024)Fast Algorithms for Minimum Homology BasisDiscrete & Computational Geometry10.1007/s00454-024-00680-8Online publication date: 24-Jul-2024
https://doi.org/10.1007/s00454-024-00680-8
Akmal SJin C(2024)An Efficient Algorithm for All-Pairs Bounded Edge ConnectivityAlgorithmica10.1007/s00453-023-01203-286:5(1623-1656)Online publication date: 22-Jan-2024
https://doi.org/10.1007/s00453-023-01203-2
Feng YWoodruff DKrause ABrunskill ECho KEngelhardt BSabato SScarlett J(2023)Improved algorithms for white-box adversarial streamsProceedings of the 40th International Conference on Machine Learning10.5555/3618408.3618807(9962-9975)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.5555/3618408.3618807
Zverkov OSeliverstov A(2023)Effective Lower Bounds on the Matrix Rank and Their ApplicationsПрограммирование10.31857/S0132347423020176(79-86)Online publication date: 1-Sep-2023
https://doi.org/10.31857/S0132347423020176
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