A stability result using the matrix norm to bound the permanent

Article
  • 8 Downloads

Abstract

We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose A is an n × n matrix over C (resp. R), and let P denote the set of n × n matrices over C (resp. R) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of A satisfies | per(A);| ≤;|;|A;|;|2 n with equality iff A/|;|A|;|2P (where |;|A|;|2 is the operator 2-norm of A). We show a stability version of this result asserting that unless A is very close (in a particular sense) to one of these extremal matrices, its permanent is exponentially smaller (as a function of n) than |;|A|;|2 n . In particular, for any fixed α, β > 0, we show that |per(A)| is exponentially smaller than |;|A|;|2 n unless all but at most αn rows contain entries of modulus at least |;|A|;|2(1 − β).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Aaronson and T. Hance, Generalizing and derandomizing Gurvits’s approximation algorithm for the permanent, Quantum Information & Computation 14 (2014), 541–559MathSciNetGoogle Scholar
  2. [2]
    S. Aaronson and H. Nguyen, Near invariance of the hypercube, Israel Journal of Mathematics 212 (2016), 385–417MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    N. Ailon and E. Liberty, Fast dimension reduction using Rademacher series on dual BCH codes, Discrete & Computational Geometry 42 (2009), 615–630MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    A. Bonami, Étude des coefficients de Fourier des fonctions de Lp(G), Université de Grenoble. Annales de l’Institut Fourier 20 (1970), 335–402 (1971).Google Scholar
  5. [5]
    D. G. Glynn, The permanent of a square matrix, European Journal of Combinatorics 31 (2010), 1887–1891MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    L. Gurvits, On the complexity of mixed discriminants and related problems, in Mathematical Foundations of Computer Science 2005, Lecture Notes in Computer Science, Vol. 3618, Springer, Berlin, 2005, pp. 2005–447.Google Scholar
  7. [7]
    L. Gurvits and A. Samorodnitsky, Bounds on the permanent and some applications, in 55th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2014, IEEE, Los Alamitos, CA, 2014, pp. 2014–90.Google Scholar
  8. [8]
    H. König, C. Schütt and N. Tomczak-Jaegermann, Projection constants of symmetric spaces and variants of Khintchine’s inequality, Journal für die Reine und Angewanste Mathematik 511 (1999), 1–42MathSciNetMATHGoogle Scholar
  9. [9]
    M. Ledoux and M. Talagrand, Probability in Banach spaces, Classics in Mathematics, Springer-Verlag, Berlin, 2011MATHGoogle Scholar
  10. [10]
    H. Nguyen, On matrices of large permanent, Private communication, 2016Google Scholar
  11. [11]
    R. O’Donnell, Analysis of Boolean Functions, Cambridge University Press, New York, 2014CrossRefMATHGoogle Scholar
  12. [12]
    W. Schudy and M. Sviridenko, Concentration and moment inequalities for polynomials of independent random variables, in Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2012, pp. 2012–437.Google Scholar
  13. [13]
    L. G. Valiant, The complexity of computing the permanent, Theoretical Computer Science 8 (1979), 189–201MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA

Personalised recommendations