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Acta Mathematica Vietnamica

, Volume 43, Issue 4, pp 701–713 | Cite as

The Permanent Functions of Tensors

  • Qing-Wen Wang
  • Fuzhen Zhang
Article
  • 71 Downloads

Abstract

By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the definitions of 1-permanent, 2-permanent, and k-permanent of a tensor in terms of hyperplanes, planes, and k-planes of the tensor; we discuss the polytopes of stochastic tensors; at the end, we present an extension of the generalized matrix function for tensors.

Keywords

Birkhoff-von Neumann theorem Doubly stochastic matrix Hypermatrix Matrix of higher order Multidimensional array Permanent Polytope Stochastic tensor Tensor 

Mathematics Subject Classification (2010)

15A15 15A02 52B12 

Notes

Acknowledgements

The work was done while the second author was visiting Shanghai University during his sabbatical leave from Nova Southeastern University. This expository article was written based on the second author’s presentation at ICMAA in Da Nang, Vietnam, June 14–18, 2017. The authors appreciate the communications with C. Bu, L. Cui, S. Hu, L. Qi, A. Taranenko, Y. Wei, and G. Yu during the preparation of the manuscript.

Funding Information

The work of Wang was partially supported by the Natural Science Foundation of China (11571220); the work of Zhang was partially supported by an NSU PFRDG Research Scholar grant.

References

  1. 1.
    Ahmed, M.: Algebraic Combinatorics of Magic Squares. University of Califorina - Davis, Ph.D. Thesis (2004)Google Scholar
  2. 2.
    Barvinok, A.: Computing the permanent of (some) complex matrices. Found. Comput. Math. 16(2), 329–342 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brualdi, R.A., Csima, J.: Stochastic patterns. J. Combin. Theory Ser. A 19, 1–12 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brualdi, R.A., Csima, J.: Extremal plane stochastic matrices of dimension three. Linear Algebra Appl. 11(2), 105–133 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brualdi, R.A., Csima, J.: Small matrices of large dimension. Proceedings of the First Conference of the International Linear Algebra Society (Provo, UT, 1989). Linear Algebra Appl. 150, 227–241 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bu, C., Wang, W., Sun, L., Zhou, J.: Minimum (maximum) rank of sign pattern tensors and sign nonsingular tensors. Linear Algebra Appl. 483, 101–114 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cayley, A.: On the theory of linear transformations. Cambridge Math. J. 4, 193–209 (1845)Google Scholar
  8. 8.
    Cayley, A.: Sur les déterminants gauches. (Suite du Memoire T. XXXII. p. 119). (French). J. Reine Angew. Math. 38, 93–96 (1849)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chang, H., Paksoy, V.E., Zhang, F.: Polytopes of stochastic tensors. Ann. Funct. Anal. 7(3), 386–393 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Che, M., Bu, C., Qi, L., Wei, Y.: Nonnegative tensors revisited: plane stochastic tensors. Linear Multilinear Algebra.  https://doi.org/10.1080/03081087.2018.1453469
  11. 11.
    Christensen, J.P.R., Fischer, P.: Multidimensional stochastic matrices and error-correcting codes. Linear Algebra Appl. 183, 255–276 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cifuentes, D., Parrilo, P.A.: An efficient tree decomposition method for permanents and mixed discriminants. Linear Algebra Appl. 493, 45–81 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Colbourn, C.J., Dinitz, J.: Handbook of Combinatorial Designs, Second Edition. Chapman and hall/CRC Press, Boca Raton (2006)Google Scholar
  14. 14.
    Cui, L.-B., Li, W., Ng, M.K.: Birkhoff–von Neumann theorem for multistochastic tensors. SIAM J. Matrix Anal. Appl. 35(3), 956–973 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ding, W., Wei, Y.: Theory and Computation of Tensors. Elsevier/Academic Press, London (2016)zbMATHGoogle Scholar
  16. 16.
    Dow, S.J., Gibson, P.M.: Permanents of d-dimensional matrices. Linear Algebra Appl. 90, 133–145 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dow, S.J., Gibson, P.M.: An upper bound for the permanent of a 3-dimensional (0,1)-matrix. Proc. Am. Math. Soc. 99(1), 29–34 (1987)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fischer, P., Swart, E.R.: Three-dimensional line stochastic matrices and extreme points. Linear Algebra Appl. 69, 179–203 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants and Multidimensional Determinants. Mathematics: Theory \(\&\) Applications. Birkhäuser Boston, Inc., Boston (1994)CrossRefzbMATHGoogle Scholar
  20. 20.
    Hu, S., Huang, Z.-H., Ling, C., Qi, L.: On determinants and eigenvalue theory of tensors. J. Symbolic Comput. 50, 508–531 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jurkat, W.B., Ryser, H.J.: Extremal configurations and decomposition theorems. I. J. Algebra 8, 194–222 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ke, R., Li, W., Xiao, M.: Characterization of extreme points of multi-stochastic tensors. Comput. Methods Appl. Math. 16(3), 459–274 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liang, Y., Ke, R., Li, W., Cui, L.: On the extreme point of m-stochastic tensors. Manuscript (2017)Google Scholar
  24. 24.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Li, Z., Zhang, F., Zhang, X.-D.: On the number of vertices of the stochastic tensor polytope. Linear Multilinear Algebra 65(10), 2064–2075 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lim, L.-H.: Tensors and Hypermatrices. Chapter 15 in Handbook of Linear Algebra, Second Edition. Chapman and hall/CRC, Boca Raton (2013)Google Scholar
  27. 27.
    Linial, N., Luria, Z.: An upper bound on the number of high-dimensional permutations. Combinatorica 34(4), 471–486 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Marchi, E., Tarazaga, P.: About \((k, n)\) stochastic matrices. Linear Algebra Appl. 26, 15–30 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Merris, R.: Trace functions. I. Duke. Math. J. 38, 527–530 (1971)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Minc, H.: Theory of permanents 1978-1981. Linear Multilinear Algebra 12(4), 227–263 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Oldenburger, R.: Higher dimensional determinants. Am. Math. Monthly 47(1), 25–33 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Qi, L., Luo, Z.: Tensor Analysis. Spectral Theory and Special Tensors. Society for Industrial and Applied Mathematics, Philadelphia (2017)CrossRefzbMATHGoogle Scholar
  33. 33.
    Schrage, G.: Some inequalities for multidimensional (0,1)-matrices. Discrete Math. 23(2), 169–175 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Shao, J.-Y., Shan, H.-Y., Zhang, L.: On some properties of the determinants of tensors. Linear Algebra Appl. 439(10), 3057–3069 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Taranenko, A.A.: Permanents of multidimensional matrices: properties and applications. (Russian) Diskretn. Anal. Issled. Oper. 23(4), 35–101 (2016). translation in J. Appl. Ind. Math. 10(4), 567–604MathSciNetzbMATHGoogle Scholar
  36. 36.
    Tichy, M.C.: Sampling of partially distinguishable bosons and the relation to the multidimensional permanent. Phys. Rev. A 022316, 91 (2015)Google Scholar
  37. 37.
    Zhang, F.: Matrix Theory: Basic Results and Techniques, Second edition. Springer, New York (2011)Google Scholar
  38. 38.
    Zhang, F.: An update on a few permanent conjectures. Spec. Matrices 4(15-02), 305–316 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Shanghai UniversityShanghaiPeople’s Republic of China
  2. 2.Nova Southeastern UniversityFort LauderdaleUSA

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