Inequalities on Generalized Tensor Functions with Diagonalizable and Symmetric Positive Definite Tensors
Abstract
The main purpose of this paper is to investigate inequalities on symmetric sums of diagonalizable and positive definite tensors. In particular, we generalize the well-known Hlawka and Popoviciu inequalities to the case of diagonalizable and positive definite tensors. As corollaries, we extend Hlawka and Popoviciu inequalities for the combinatorial determinant, permanent and immanant of tensors, and generalized tensor functions.References
S. Avgustinovich, Multidimensional permanents in enumeration problems, Journal of Applied and Industrial Mathematics, 4 (2010), pp. 19-20.
A. Barvinok, Computing the permanent of (some) complex matrices, Foundations of Computational Mathematics, 16 (2016), pp. 329-342.
W. Berndt and S. Sra, Hlawka-popoviciu inequalities on positive definite tensors, Linear Algebra and its Applications, 486 (2015), pp. 317-327.
A. Cayley, On the theory of determinants, Transaction of the Cambridge Philosophical Society, 8 (1843), pp. 1-16.
X. Chang, V. E. Paksoy, and F. Zhang, An inequality for tensor product of positive operators and its applications, Linear Algebra and its Applications, 498 (2014), pp. 99-105.
M. Che, C. Bu, L. Qi, and Y. Wei, Nonnegative tensors revisited: Plane stochastic tensors, Linear and Multilinear Algebra, Available online (2018). URL:https://doi.org/10.1080/03081087.2018. 1453469.
M. Che, L. Qi, and Y. Wei, Positive-definite tensors to nonlinear complementarity problems, Journal of Optimization Theory and Applications, 168 (2016), pp. 475-487.
H. Chen, G. Li, and L. Qi, SOS tensor decomposition: Theory and applications, Communications in Mathematical Sciences, 14 (2016), pp. 2073-2100.
W. Ding, L. Qi, and Y. Wei, Inheritance properties and sum-of-squares decomposition of hankel tensors: Theory and algorithms, BIT Numerical Mathematics, 57 (2017), pp. 169-190.
S. J. Dow and P. M. Gibson, Permanents of d dimensional matrices, Linear Algebra and its Applications, 90 (1987), pp. 133-145.
J. Fan and A. Zhou, A semidefinite algorithm for completely positive tensor decomposition, Computational Optimization and Applications, 66 (2017), pp. 267-283.
W. Fechner, Hlawkas functional inequality, Aequationes Mathematicae, 87 (2014), pp. 71-87.
S. Hu, G. Li, and L. Qi, A tensor analogy of Yuan’s theorem of the alternative and polynomial optimization with sign structure, Journal of Optimization Theory and Applications, 168 (2016), pp. 446-474.
S. Huang, C. Li, Y. Poon, and Q. Wang, Inequalities on generalized matrix functions, Linear and Multilinear Algebra, Available online (2017). URL:https://doi.org/10.1080/03081087.2016. 1239690.
T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455-500.
N. Lee and A. Cichockia, Fundamental tensor operations for large-scale data analysis in tensor train formats, arXiv preprint arXiv:1405.7786v2, (2016).
G. Li, L. Qi, and Y. Xu, SOS-Hankel tensors: Theory and application, arXiv: Spectral Theory, (2014).
L.-H. Lim, Tensors and hypermatrices. Chapter 15 in Handbook of Linear Algebra, CRC Press, Boca Raton, FL, 2 ed. 2013.
Z. Luo and L. Qi, Completely positive tensors: properties, easily checkable subclasses, and tractable relaxations, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1675-1698.
M. Marcus, Finite Dimensional Multilinear Algebra. Part II, vol. 23 of Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1975.
M. Marcus and H. Minc, Generalized matrix functions, Transactions of the American Mathematical Society, 116 (1965), pp. 316-329.
R. Merris, Multilinear Algebra, vol. 8 of Algebra, Logic and Applications, Gordon and Breach Science Publishers, Amsterdam, 1997.
R. Oldenburger, Higher dimensional determinants, The American Mathematical Monthly, 47 (1940), pp. 25-33.
V. E. Paksoy, R. Turkmen, and F. Zhang, Inequalities of generalized matrix functions via tensor products, Electronic Journal of Linear Algebra, 27 (2014), pp. 332-341.
L. Qi, Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), pp. 1302-1324.
L. Qi, C. Xu, and Y. Xu, Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 1227-1241.
L. H. Rice, Introduction to higher determinants, Journal of Mathematics and Physics, 9 (1930), pp. 47-70.
A. A. Taranenko, Multidimensional permanents and an upper bound on the number of transversals in Latin squares, Journal of Combinatorial Designs, 23 (2015), pp. 305-320.
, Permanents of multidimensional matrices: Properties and applications, Journal of Applied and Industrial Mathematics, 10 (2016), pp. 567-604.
C. Xu, Z. Chen, and L. Qi, On {0, 1} CP tensors and CP pseudographs, Linear Algebra Appl., 557 (2018), pp. 287-306.
C. Xu, Z. Luo, L. Qi, and Z. Chen, {0, 1} completely positive tensors and multi-hypergraphs, Linear Algebra Appl., 510 (2016), pp. 110-123.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
