Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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BEYOND THE CACTUS RANK OF TENSORS

  • Received : 2017.10.25
  • Accepted : 2018.02.01
  • Published : 2018.09.30
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Abstract

We study additive decompositions (and generalized additive decompositions with a zero-dimensional scheme instead of a finite sum of rank 1 tensors), which are not of minimal degree (for sums of rank 1 tensors with more terms than the rank of the tensor, for a zero-dimensional scheme a degree higher than the cactus rank of the tensor). We prove their existence for all degrees higher than the rank of the tensor and, with strong assumptions, higher than the cactus rank of the tensor. Examples show that additional assumptions are needed to get the minimally spanning scheme of degree cactus +1.

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References

  1. E. Ballico, Tensor ranks on tangent developable of Segre varieties, Linear Multilinear Algebra 61 (2013), no. 7, 881-894. https://doi.org/10.1080/03081087.2012.716430
  2. E. Ballico and A. Bernardi, Stratification of the fourth secant variety of Veronese varieties via the symmetric rank, Adv. Pure Appl. Math. 4 (2013), no. 2, 215-250.
  3. E. Ballico and A. Bernard, Curvilinear schemes and maximum rank of forms, Matematiche (Catania) 72 (2017), no. 1, 137-144.
  4. E. Ballico and A. Bernard, On the ranks of the third secant variety of Segre-Veronese embeddings, arXiv: 1701.06845.
  5. E. Ballico, A. Bernardi, and L. Chiantini, On the dimension of contact loci and the identifiability of tensors, arXiv: 1706.02746.
  6. E. Ballico, A. Bernardi, L. Chiantini, and E. Guardo, Bounds on the tensor rank, arXiv: 1705.02299.
  7. A. Bernardi, J. Brachat, and B. Mourrain, A comparison of different notions of ranks of symmetric tensors, Linear Algebra Appl. 460 (2014), 205-230. https://doi.org/10.1016/j.laa.2014.07.036
  8. A. Bernardi and K. Ranestad, On the cactus rank of cubics forms, J. Symbolic Comput. 50 (2013), 291-297. https://doi.org/10.1016/j.jsc.2012.08.001
  9. C. Bocci and L. Chiantini, On the identifiability of binary Segre products, J. Algebraic Geom. 22 (2013), no. 1, 1-11. https://doi.org/10.1090/S1056-3911-2011-00592-4
  10. C. Bocci, L. Chiantini, and G. Ottaviani, Refined methods for the identifiability of tensors, Ann. Mat. Pura Appl. (4) 193 (2014), no. 6, 1691-1702. https://doi.org/10.1007/s10231-013-0352-8
  11. W. Buczynska and J. Buczynski, Secant varieties to high degree veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes, J. Algebraic Geom. 23 (2014), 63-90.
  12. J. Buczynski and J. M. Landsberg, On the third secant variety, J. Algebraic Combin. 40 (2014), no. 2, 475-502. https://doi.org/10.1007/s10801-013-0495-0
  13. M. V. Catalisano, L. Chiantini, A. V. Geramita, and A. Oneto Waring-like decompositions of polynomials, 1, Linear Algebra Appl. 533 (2017), 311-325. https://doi.org/10.1016/j.laa.2017.07.021
  14. L. Chiantini and C. Ciliberto, On the concept of k-secant order of a variety, J. London Math. Soc. (2) 73 (2006), no. 2, 436-454. https://doi.org/10.1112/S0024610706022630
  15. L. Chiantini and C. Ciliberto, On the dimension of secant varieties, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1267-1291.
  16. L. Chiantini, M. Mella, and G. Ottaviani, One example of general unidentifiable tensors, J. Algebr. Stat. 5 (2014), no. 1, 64-71.
  17. L. Chiantini and G. Ottaviani, On generic identifiability of 3-tensors of small rank, SIAM J. Matrix Anal. Appl. 33 (2012), no. 3, 1018-1037. https://doi.org/10.1137/110829180
  18. L. Chiantini, G. Ottaviani, and N. Vannieuwenhoven, An algorithm for generic and low-rank specific identifiability of complex tensors, SIAM J. Matrix Anal. Appl. 35 (2014), no. 4, 1265-1287. https://doi.org/10.1137/140961389
  19. L. Chiantini, G. Ottaviani, and N. Vannieuwenhoven, On generic identifiability of symmetric tensors of subgeneric rank, Trans. Amer. Math. Soc. 369 (2017), no. 6, 4021-4042.
  20. L. Chiantini, G. Ottaviani, and N. Vannieuwenhoven, Effective criteria for specific identifiability of tensors and forms, SIAM J. Ma- trix Anal. Appl. 38 (2017), no. 2, 656-681. https://doi.org/10.1137/16M1090132
  21. H. Derksen, Kruskal's uniqueness inequality is sharp, Linear Algebra Appl. 438 (2013), no. 2, 708-712. https://doi.org/10.1016/j.laa.2011.05.041
  22. I. Domanov and L. De Lathauwer, On the uniqueness of the canonical polyadic decomposition of third-order tensors - Part I: Basic results and uniqueness of one factor matrix, SIAM J. Matrix Anal. Appl. 34 (2013), no. 3, 855-875. https://doi.org/10.1137/120877234
  23. I. Domanov and L. De Lathauwer, On the uniqueness of the canonical polyadic decomposition of third-order tensors - Part II: Uniqueness of the overall decomposition, SIAM J. Matrix Anal. Appl. 34 (2013), no. 3, 876-903. https://doi.org/10.1137/120877258
  24. I. Domanov and L. De Lathauwer, Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL, SIAM J. Matrix Anal. Appl. 36 (2015), no. 4, 1567-1589. https://doi.org/10.1137/140970276
  25. F. Galuppi and M. Mella, Identifiability of homogeneous polynomials and Cremona Transformations, Preprint arXiv:1606.06895, 2016; To appear in J. Reine Angew. Math.; DOI: https://doi.org/10.1515/crelle-2017-0043.
  26. A. Iarrobino and V. Kanev, Power Sums, Gorenstein Algebras, and Determinantal Loci, Lecture Notes in Mathematics, 1721, Springer-Verlag, Berlin, 1999.
  27. J. B. Kruskal, Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics, Linear Algebra and Appl. 18 (1977), no. 2, 95-138. https://doi.org/10.1016/0024-3795(77)90069-6
  28. J. M. Landsberg, Tensors: geometry and applications, Graduate Studies in Mathematics, 128, American Mathematical Society, Providence, RI, 2012.
  29. M. Mella, Singularities of linear systems and the Waring problem, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5523-5538. https://doi.org/10.1090/S0002-9947-06-03893-1
  30. K. Ranestad and F.-O. Schreyer, On the rank of a symmetric form, J. Algebra 346 (2011), 340-342. https://doi.org/10.1016/j.jalgebra.2011.07.032
  31. N. D. Sidiropoulos and R. Bro, On the uniqueness of multilinear decomposition of N-way arrays, J. Chemometrics 14 (2000), 229-239. https://doi.org/10.1002/1099-128X(200005/06)14:3<229::AID-CEM587>3.0.CO;2-N