Duke Mathematical Journal

Bounded-rank tensors are defined in bounded degree

Jan Draisma and Jochen Kuttler

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Matrices of rank at most k are defined by the vanishing of polynomials of degree k+1 in their entries (namely, their ((k+1)×(k+1))-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this statement for tensors of arbitrary dimension, where matrices correspond to two-dimensional tensors. More specifically, we prove that for each k there exists an upper bound d=d(k) such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d, regardless of the dimension of the tensor and regardless of its size in each dimension. Our proof involves passing to an infinite-dimensional limit of tensor powers of a vector space, whose elements we dub infinite-dimensional tensors, and exploiting the symmetries of this limit in crucial ways.

Article information

Source
Duke Math. J., Volume 163, Number 1 (2014), 35-63.

Dates
First available in Project Euclid: 8 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1389190324

Digital Object Identifier
doi:10.1215/00127094-2405170

Mathematical Reviews number (MathSciNet)
MR3161311

Zentralblatt MATH identifier
1314.14109

Subjects
Primary: 14Q15: Higher-dimensional varieties
Secondary: 15A69: Multilinear algebra, tensor products 13E05: Noetherian rings and modules

Citation

Draisma, Jan; Kuttler, Jochen. Bounded-rank tensors are defined in bounded degree. Duke Math. J. 163 (2014), no. 1, 35--63. doi:10.1215/00127094-2405170. https://projecteuclid.org/euclid.dmj/1389190324


Export citation

References

  • [1] E. S. Allman, C. Matias, and J. A. Rhodes, Identifiability of parameters in latent structure models with many observed variables, Ann. Statist. 37 (2009), 3099–3132.
  • [2] E. S. Allman and J. A. Rhodes, Phylogenetic ideals and varieties for the general Markov model, Adv. in Appl. Math. 40 (2008), 127–148.
  • [3] E. S. Allman, J. A. Rhodes, and A. Taylor, A semialgebraic description of the general Markov model on phylogenetic trees, preprint, arXiv:1212.1200v1 [q-bio.PE].
  • [4] D. J. Bates and L. Oeding, Toward a salmon conjecture, Exp. Math. 20 (2011), 358–370.
  • [5] A. Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.
  • [6] A. E. Brouwer and J. Draisma, Equivariant Gröbner bases and the Gaussian two-factor model, Math. Comp. 80 (2011), 1123–1133.
  • [7] P. Bürgisser, M. Clausen, and M. A. Shokrollahi, Algebraic Complexity Theory, Grundlehren Math. Wiss. 315, Springer, Berlin, 1997.
  • [8] P. Bürgisser, J. M. Landsberg, L. Manivel, and J. Weyman, An overview of mathematical issues arising in the geometric complexity theory approach to $\textit{VP}\neq\textit{VNP}$, SIAM J. Comput. 40 (2011), 1179–1209.
  • [9] M. Casanellas and J. Fernández-Sánchez, Relevant phylogenetic invariants of evolutionary models, J. Math. Pures Appl. (9) 96 (2011), 207–229.
  • [10] M. Casanellas and S. Sullivant, “The strand symmetric model” in Algebraic Statistics for Computational Biology, Cambridge Univ. Press, Cambridge, 2005, 305–321.
  • [11] M. V. Catalisano, A. V. Geramita, and A. Gimigliano, “On the rank of tensors, via secant varieties and fat points” in Zero-Dimensional Schemes and Applications (Naples, 2000), Queen’s Papers in Pure and Appl. Math. 123, Queen’s Univ., Kingston, Ont., 2002, 133–147.
  • [12] Maria V. Catalisano, Anthony V. Geramita, and Alessandro Gimigliano, Higher secant varieties of the Segre varieties $\mathbb{P}^{1}\times\dots\times\mathbb{P}^{1}$, J. Pure Appl. Algebra 201 (2005), 367–380.
  • [13] D. E. Cohen, On the laws of a metabelian variety, J. Algebra 5 (1967), 267–273.
  • [14] V. de Silva and L.-H. Lim, Tensor rank and the ill-posedness of the best low-rank approximation problem, SIAM J. Matrix Anal. Appl. 30 (2008), 1084–1127.
  • [15] J. Draisma, Finiteness for the $k$-factor model and chirality varieties, Adv. Math. 223 (2010), 243–256.
  • [16] J. Draisma, E. Kushilevitz, and E. Weinreb, Partition arguments in multiparty communication complexity, Theoret. Comput. Sci. 412 (2011), 2611–2622.
  • [17] J. Draisma and J. Kuttler, On the ideals of equivariant tree models, Math. Ann. 344 (2009), 619–644.
  • [18] S. Friedland and E. Gross, A proof of the set-theoretic version of the salmon conjecture, J. Algebra 356 (2012), 374–379.
  • [19] L. D. Garcia, M. Stillman, and B. Sturmfels, Algebraic geometry of Bayesian networks, J. Symbolic Comput. 39 (2005), 331–355.
  • [20] G. Higman, Ordering by divisibility in abstract algebras, Proc. Lond. Math. Soc. (3) 2 (1952), 326–336.
  • [21] C. J. Hillar and L.-H. Lim, Most tensor problems are NP hard, preprint, arXiv:0911.1393v5 [cs.CC].
  • [22] C. J. Hillar and S. Sullivant, Finite Gröbner bases in infinite dimensional polynomial rings and applications, Adv. Math. 229 (2012), 1–25.
  • [23] J. B. Kruskal, The theory of well-quasi-ordering: A frequently discovered concept, J. Combin. Theory Ser. A 13 (1972), 297–305.
  • [24] J. M. Landsberg, The border rank of the multiplication of $2\times2$ matrices is seven, J. Amer. Math. Soc. 19 (2006), 447–459.
  • [25] J. M. Landsberg, Geometry and the complexity of matrix multiplication, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 247–284.
  • [26] J. M. Landsberg, $P$ versus $NP$ geometry, J. Symbolic Comput. 45 (2010), 1369–1377.
  • [27] J. M. Landsberg and L. Manivel, On the ideals of secant varieties of Segre varieties, Found. Comput. Math. 4 (2004), 397–422.
  • [28] J. M. Landsberg and J. Weyman, On the ideals and singularities of secant varieties of Segre varieties, Bull. Lond. Math. Soc. 39 (2007), 685–697.
  • [29] C. Raicu, Secant varieties of Segre–Veronese varieties, Ph.D. dissertation, University of California, Berkeley, Berkeley, Calif., 2011.
  • [30] A. Snowden, Syzygies of Segre embeddings and $\delta$-modules, Duke Math. J. 162 (2013), 225–277.
  • [31] B. Sturmfels and S. Sullivant, Toric ideals of phylogenetic invariants, J. Comput. Biol. 12 (2005), 204–228.
  • [32] V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969), 354–356.
  • [33] V. Strassen, Rank and optimal computation of generic tensors, Linear Algebra Appl. 52/53 (1983), 645–685.