Document Open Access Logo

Singularity of Random Integer Matrices with Large Entries

Authors Sankeerth Rao Karingula , Shachar Lovett

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2021.33.pdf
  • Filesize: 0.68 MB
  • 16 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Coding Theory
  • Random matrix theory
  • Singularity probability MDS codes
  • Error correction codes
  • Littlewood Offord
  • Fourier Analysis

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We study the singularity probability of random integer matrices. Concretely, the probability that a random n × n matrix, with integer entries chosen uniformly from {-m,…,m}, is singular. This problem has been well studied in two regimes: large n and constant m; or large m and constant n. In this paper, we extend previous techniques to handle the regime where both n,m are large. We show that the probability that such a matrix is singular is m^{-cn} for some absolute constant c > 0. We also provide some connections of our result to coding theory.

Cite As Get BibTex

Sankeerth Rao Karingula and Shachar Lovett. Singularity of Random Integer Matrices with Large Entries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.33

Author Details

Sankeerth Rao Karingula
  • Department of Computer Science, University of California San Diego, CA, USA
Shachar Lovett
  • Department of Computer Science, University of California San Diego, CA, USA

Funding

  • Karingula, Sankeerth Rao: Research supported by NSF CCF award 1614023.
  • Lovett, Shachar: Research supported by NSF CCF award 1614023.

Acknowledgements

We would like to thank Roman Vershynin and Konstantin Tikhomirov for helpful discussions.

References

  1. Simeon Ball. On sets of vectors of a finite vector space in which every subset of basis size is a basis. Journal of the European Mathematical Society, 14(3):733-748, 2012. Google Scholar
  2. Jean Bourgain, Van H Vu, and Philip Matchett Wood. On the singularity probability of discrete random matrices. Journal of Functional Analysis, 258(2):559-603, 2010. Google Scholar
  3. Anthony Carbery and James Wright. Distributional and L^q norm inequalities for polynomials over convex bodies in ℝⁿ. Mathematical research letters, 8(3):233-248, 2001. Google Scholar
  4. CG Esseen. On the Kolmogorov-Rogozin inequality for the concentration function. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 5(3):210-216, 1966. Google Scholar
  5. Jeff Kahn, János Komlós, and Endre Szemerédi. On the probability that a random ± 1-matrix is singular. Journal of the American Mathematical Society, 8(1):223-240, 1995. Google Scholar
  6. Yonathan Katznelson. Singular matrices and a uniform bound for congruence groups of SL_n(ℤ). Duke Mathematical Journal, 69(1):121-136, 1993. Google Scholar
  7. János Komlós. On determinant of (0, 1) matrices. Studia Science Mathematics Hungarica, 2:7-21, 1967. Google Scholar
  8. L Leindler. On a certain converse of Hölder’s inequality. In Proceedings of the 1971 Oberwolfach Conference, BirkhHuser Verlag. Basel-Stuttgart, 1972. Google Scholar
  9. Vitali D Milman and Gideon Schechtman. Asymptotic theory of finite dimensional normed spaces: Isoperimetric inequalities in riemannian manifolds, volume 1200. Springer, 2009. Google Scholar
  10. András Prékopa. On logarithmic concave measures and functions. Acta Scientiarum Mathematicarum, 34:335-343, 1973. Google Scholar
  11. Mark Rudelson. Lecture notes on non-asymptotic theory of random matrices, 2013. Google Scholar
  12. Mark Rudelson and Roman Vershynin. The Littlewood-Offord problem and invertibility of random matrices. Advances in Mathematics, 218(2):600-633, 2008. Google Scholar
  13. Mark Rudelson and Roman Vershynin. Non-asymptotic theory of random matrices: extreme singular values. In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II-IV: Invited Lectures, pages 1576-1602. World Scientific, 2010. Google Scholar
  14. Jacob T Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM (JACM), 27(4):701-717, 1980. Google Scholar
  15. Beniamino Segre. Curve razionali normali ek-archi negli spazi finiti. Annali di Matematica Pura ed Applicata, 39(1):357-379, 1955. Google Scholar
  16. Richard Singleton. Maximum distance q-nary codes. IEEE Transactions on Information Theory, 10(2):116-118, 1964. Google Scholar
  17. Terence Tao and Van Vu. On random ± 1 matrices: singularity and determinant. Random Structures & Algorithms, 28(1):1-23, 2006. Google Scholar
  18. Terence Tao and Van Vu. On the singularity probability of random Bernoulli matrices. Journal of the American Mathematical Society, 20(3):603-628, 2007. Google Scholar
  19. Konstantin Tikhomirov. Singularity of random Bernoulli matrices. Annals of Mathematics, 191(2):593-634, 2020. Google Scholar
  20. Santosh S Vempala, Ruosong Wang, and David P Woodruff. The communication complexity of optimization. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1733-1752. SIAM, 2020. Google Scholar
  21. Richard Zippel. Probabilistic algorithms for sparse polynomials. In International symposium on symbolic and algebraic manipulation, pages 216-226. Springer, 1979. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact