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Plane Stochastic Tensors

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Theory and Computation of Complex Tensors and its Applications

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Abstract

We study combinatorial properties of nonnegative tensors. We make the following contributions: (1) we obtain equivalent conditions for sign nonsingular tensors and relationships between the combinatorial determinant and the permanent of nonnegative tensors, in Theorems 5.2.1 and 5.2.2; (2) the sets of plane stochastic tensors and totally plane stochastic tensors are closed, bounded and convex sets, and an nonnegative tensor has a plane stochastic pattern if and only if its positive entries are contained in a positive diagonal, in Lemma 5.3.1 and Theorem 5.3.2; (3) from a nonnegative tensor, we propose a normalization algorithm which converges to a plane stochastic tensor, in Theorem 5.3.8; (4) we discuss the boundlessness of the diagonal products of any nonnegative tensor and obtain a probabilistic algorithm after Theorem 5.4.4 for locating a positive diagonal in a (0, 1)-tensor; (5) we explore the axial N-index assignment problem via the set of plane stochastic tensors in Sect. 5.5.

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Notes

  1. 1.

    A tensor is a (0,  1)-tensor [1], if its entries are chosen from the set {0,  1}.

  2. 2.

    For any \(\mathcal {A}\in T_{N,I}\) and a given n, the mode-n (i 1, …, i n−1, i n+1, …, i N)-fiber [63] of \(\mathcal {A}\) is defined by \(\mathcal {A}(i_1,\dots ,i_{n-1},:,i_{n+1},\dots ,i_N)\in \mathbb {R}^I\) for all i 1, …, i n−1, i n+1, …, i N.

References

  1. S. Dow, P. Gibson, Permanents of d-dimensional matrices. Linear Algebra Appl. 90, 133–145 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Brualdi, J. Csima, Extremal plane stochastic matrices of dimension three. Linear Algebra Appl. 11(2), 105–133 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Csima, Multidimensional stochastic matrices and patterns. J. Algebra 14(2), 194–202 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. W. Jurkat, H. Ryser, Extremal configurations and decomposition theorems. I. J. Algebra 8(2), 194–222 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Lu, K. Plataniotis, A. Venetsanopoulos, Multilinear Subspace Learning: Dimensionality Reduction of Multidimensional Data (CRC, Boca Raton, 2013)

    Book  Google Scholar 

  6. A. Berman, R. Plemmons, Nonnegative Matrices in the Mathematical Sciences (Society for Industrial and Applied Mathematics, Philadelphia, 1994)

    Book  MATH  Google Scholar 

  7. R. Brualdi, P. Gibson, Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function. J. Comb. Theor. A 22(2), 194–230 (1977)

    MATH  Google Scholar 

  8. Q. Yang, Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors II. SIAM J. Matrix Anal. Appl. 32(4), 1236–1250 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. K. Chang, K. Pearson, T. Zhang, Perron-Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6(2), 507–520 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Brualdi, J. Csima, Stochastic patterns. J. Comb. Theor. A 19, 1–12 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Cui, W. Li, M. Ng, Birkhoff–von Neumann theorem for multistochastic tensors. SIAM J. Matrix Anal. Appl. 35(3), 956–973 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Raftery, A model for high-order markov chains. J. R. Stat. Soc. Ser. B 47, 528–539 (1985)

    MathSciNet  MATH  Google Scholar 

  13. K. Chang, T. Zhang, On the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors. J. Math. Anal. Appl. 408(2), 525–540 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. K. Chang, L. Qi, T. Zhang, A survey on the spectral theory of nonnegative tensors. Numer. Linear Algebra Appl. 20(6), 891–912 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Gleich, L. Lim, Y. Yu, Multilinear PageRank. SIAM J. Matrix Anal. Appl. 36(4), 1507–1541 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Hu, L. Qi, Convergence of a second order Markov chain. Appl. Math. Comput. 241, 183–192 (2014)

    MathSciNet  MATH  Google Scholar 

  17. J. Christensen, P. Fischer, Multidimensional stochastic matrices and error-correcting codes. Linear Algebra Appl. 183, 255–276 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. V. Pless, Introduction to the Theory of Error-correcting Codes (Wiley, New York, 1998)

    Book  MATH  Google Scholar 

  19. G. Golub, C. Van Loan, Matrix Computations, 4th edn. (Johns Hopkins University Press, Baltimore, 2013)

    MATH  Google Scholar 

  20. S. Hu, Z. Huang, C. Ling, L. Qi, On determinants and eigenvalue theory of tensors. J. Symb. Comput. 50, 508–531 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. L. Qi, Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Rice, Introduction to higher determinants. J. Math. Phys. 9(1), 47–70 (1930)

    Article  MATH  Google Scholar 

  23. J. Rotman, An Introduction to the Theory of Groups, 4th edn. (Springer, New York, 1995)

    Book  MATH  Google Scholar 

  24. A. Cayley, On the theory of determinants. Trans. Camb. Philos. Soc. 8, 1–16 (1843)

    MATH  Google Scholar 

  25. R. Oldenburger, Higher dimensional determinants. Am. Math. Mon. 47(1), 25–33 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Vein, P. Dale, Determinants and their Applications in Mathematical Physics (Springer, New York, 1999)

    MATH  Google Scholar 

  27. L. Lim, Tensors and hypermatrices, in Handbook of Linear Algebra, 2nd edn. (CRC Press, Boca Raton, 2013)

    Google Scholar 

  28. L. Lim, Eigenvalues of Tensors and Spectral Hypergraph Theory (Talk) (University of California, Berkeley, 2008)

    Google Scholar 

  29. K. Pearson, T. Zhang, On spectral hypergraph theory of the adjacency tensor. Graphs and Combinatorics 30(5), 1233–1248 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. C. Berge, Graphs and Hypergraphs, in North-Holland Mathematical Library, vol. 45 (North-Holland, Amsterdam, 1976)

    Google Scholar 

  31. A. Taranenko, Permanents of multidimensional matrices: Properties and applications. J. Appl. Ind. Math. 10(4), 567–604 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Barvinok, Computing the permanent of (some) complex matrices. Found. Comput. Math. 16(2), 329–342 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Avgustinovich, Multidimensional permanents in enumeration problems. J. Appl. Ind. Math. 4(1), 19–20 (2010)

    Article  MathSciNet  Google Scholar 

  34. A. Taranenko, Multidimensional permanents and an upper bound on the number of transversals in Latin squares. J. Comb. Des. 23(7), 305–320 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. D. Littlewood, The Theory of Group Characters and Matrix Representations of Groups (Oxford University Press, Oxford, 1940)

    MATH  Google Scholar 

  36. M. Marcus, Finite dimensional multilinear algebra. Part II, in Pure and Applied Mathematics, vol. 23 (Marcel Dekker, Inc., New York, 1975)

    Google Scholar 

  37. R. Merris, Multilinear algebra, in Algebra, Logic and Applications, vol. 8 (Gordon and Breach Science, Amsterdam, 1997)

    Book  MATH  Google Scholar 

  38. M. Marcus, H. Minc, Generalized matrix functions. Trans. Am. Math. Soc. 116, 316–329 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  39. W. Berndt, S. Sra, Hlawka-popoviciu inequalities on positive definite tensors. Linear Algebra Appl. 486, 317–327 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. S. Huang, C. Li, Y. Poon, Q. Wang, Inequalities on generalized matrix functions. Linear Multilinear Algebra 65, 1947–1961 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. X. Chang, V. Paksoy, F. Zhang, An inequality for tensor product of positive operators and its applications. Linear Algebra Appl. 498, 99–105 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. V. Paksoy, R. Turkmen, F. Zhang, Inequalities of generalized matrix functions via tensor products. Electron. J. Linear Algebr. 27(1), 332–341 (2014)

    MathSciNet  MATH  Google Scholar 

  43. N. Lee, A. Cichockia, Fundamental Tensor Operations for Large-Scale Data Analysis in Tensor Train Formats (2016). ArXiv preprint:1405.7786v2

    Google Scholar 

  44. M. Che, D.S. Cvetković, Y. Wei, Inequalities on generalized tensor functions associated to the diagonalizable symmetric tensors. Stat. Optim. Infor. Comput. 6(4), 483–496 (2018)

    Google Scholar 

  45. C. Bu, W. Wang, L. Sun, J. Zhou, Minimum (maximum) rank of sign pattern tensors and sign nonsingular tensors. Linear Algebra Appl. 483, 101–114 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Brualdi, B. Shader, Matrices of Sign-solvable Linear System (Cambridge University, Cambridge, 1995)

    Book  MATH  Google Scholar 

  47. R. Bapat, T. Raghavan, Nonnegative Matrices and Applications (Cambridge University, Cambridge, 1997)

    Book  MATH  Google Scholar 

  48. R. Brualdi, J. Csima, Small matrices of large dimension. Linear Algebra Appl. 150, 227–241 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  49. H. Chang, V. Paksoy, F. Zhang, Polytopes of stochastic tensors. Ann. Funct. Anal. 7(3), 386–393 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  50. P. Fischer, E. Swart, Three dimensional line stochastic matrices and extreme points. Linear Algebra Appl. 69, 179–203 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  51. Z. Li, F. Zhang, X. Zhang, On the number of vertices of the stochastic tensor polytope. Linear Multilinear Algebra 65, 2064–2075 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  52. E. Marchi, P. Tarazaga, About (k, n) stochastic matrices. Linear Algebra Appl. 26, 15–30 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  53. G. Schrage, Some inequalities for multidimensional (0,1)- matrices. Discret. Math. 23(2), 169–175 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  54. Q. Wang, F. Zhang, The permanent function of tensors. Acta Math. Vietnam 43, 701–713 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  55. R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Stat. 35, 876–879 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  56. A. Marshall, I. Olkin, Scaling of matrices to achieve specified row and column sums. Numer. Math. 12, 83–90 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  57. R. Sinkhorn, P. Knopp, Concerning nonnegative matrices and doubly stochastic matrices. Pac. J. Math. 21, 343–348 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  58. R. Bapat, D 1AD 2 theorems for multidimensional matrices. Linear Algebra Appl. 48, 437–442 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  59. T. Raghavan, On pairs of multidimensional matrices. Linear Algebra Appl. 62, 263–268 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  60. J. Franklin, J. Lorenz, On the scaling of multidimensional matrices. Linear Algebra Appl. 114/115, 717–735 (1989)

    Google Scholar 

  61. A. Shashua, R. Zass, T. Hazan, Multi-way clustering using super-symmetric non-negative tensor factorization. Lect. Notes Comput. Sci. 3954, 595–608 (2006)

    Article  Google Scholar 

  62. L. Qi, The best rank-one approximation ratio of a tensor space. SIAM J. Matrix Anal. Appl. 32(2), 430–442 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  63. T. Kolda, B. Bader, Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  64. R. Burkard, M. Dell’Amico, S. Martello, Assignment Problems (Society for Industrial and Applied Mathematics, Philadelphia, 2009)

    Book  MATH  Google Scholar 

  65. W. Pierskalla, The multidimensional assignment problem. Oper. Res. 16(2), 422–431 (1968)

    Article  MATH  Google Scholar 

  66. E. Schell, Distribution of a product by several properties, in Proceedings of the Second Symposium in Linear Programming, vol. 615, pp. 642 (1955)

    Google Scholar 

  67. W. Pierskalla, The tri-substitution method for the three-dimensional assignment problem. CORS J. 5, 71–81 (1967)

    MATH  Google Scholar 

  68. L. Qi, D. Sun, Polyhedral methods for solving three index assignment problems, in Nonlinear Assignment Problems. Combinatorial Optimization, vol. 7 (Kluwer Academic, Dordrecht, 2000), pp. 91–107

    Google Scholar 

  69. R. Malhotra, H. Bhatia, M. Puri, The three-dimensional bottleneck assignment problem and its variants. Optimization 16(2), 245–256 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  70. S. Geetha, M. Vartak, The three-dimensional bottleneck assignment problem with capacity constraints. Eur. J. Oper. Res. 73(3), 562–568 (1994)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Economics Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan, China

    Maolin Che

  2. School of Mathematical Sciences, Fudan University, Shanghai, China

    Yimin Wei

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  1. Maolin Che
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  2. Yimin Wei
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Che, M., Wei, Y. (2020). Plane Stochastic Tensors. In: Theory and Computation of Complex Tensors and its Applications. Springer, Singapore. https://doi.org/10.1007/978-981-15-2059-4_5

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