The Complexity of Tensor Rank

Abstract

We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by Håstad (J. Algorithm. 11(4), 644–654 1990). The hardness proof also implies an algebraic universality result.

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Fig. 1

Notes

  1. 1.

    The complexity class \(\exists \mathbb {R}\) was introduced explicitly in [24, 26] and some other papers, but other researchers probably thought of \(\text {ETh}(\mathbb {R})\) as a complexity class before, e.g., Shor [28], and Buss, Frandsen and Shallit [8].

  2. 2.

    We are not aware of any stronger lower bounds on \(\exists \mathbb {F}\) for any field \(\mathbb {F}\). If we allow rings, then \(\exists \mathbb {Z}\), for example, is undecidable, its complexity equivalent to the halting problem \(\emptyset ^{\prime }\). This was shown in a famous series of results by Davis, Robinson, and Matiyasevic [10, 19].

  3. 3.

    Koiran’s result assumes the generalized Riemann hypothesis (GRH); as far as we know there is no unconditional upper bound on \({\exists \mathbb {C}}\) other than PSPACE.

  4. 4.

    If \({\mathbb {Z}}\) had an existential definition in \({\mathbb {Q}}\), then it would follow that \(\exists \mathbb {Q} \equiv \exists \mathbb {Z} \equiv \emptyset ^{\prime }\). Koenigmsann [15] gives some evidence that there is no such definition (implying that his universal definition of \({\mathbb {Z}}\) in \({\mathbb {Q}}\) is optimal), however, there may be other routes towards the undecidability of \(\exists \mathbb {Q}\), and it may be undecidable without being as hard as \(\emptyset ^{\prime }\).

  5. 5.

    In other models, e.g., the Blum-Shub-Smale model [7] this was well-known earlier.

  6. 6.

    There are other definitions of eigenvalues for tensors as well.

  7. 7.

    The proof in [25] yields a quartic systems, but that can be reduced to quadratic, by removing the final (unnecessary) squaring operation.

  8. 8.

    There is also a notion of minrank for matrices with entries in {+, −}. Given such a matrix is there a real matrix of rank at most 3 with that sign pattern? This problem turns out to be \(\exists \mathbb {R}\)-hard as well [2, 3], but does not seem to be related to our minrank problem.

References

  1. 1.

    Allender, E., Burgisser, P., Kjeldgaard-Pedersen, J., Bro Miltersen, P.: On the complexity of numerical analysis. In: CCC ’06: Proceedings of the 21st Annual IEEE Conference on Computational Complexity, pp. 331–339. IEEE Computer Society, DC, USA (2006)

  2. 2.

    Basri, R., Felzenszwalb, P.F., Girshick, R.B., Jacobs, D.W., Klivans, C.J.: Visibility constraints on features of 3D objects. In: CVPR, pp. 1231–1238. IEEE Computer Society (2009)

  3. 3.

    Bhangale, A., Kopparty, S.: The complexity of computing the minimum rank of a sign pattern matrix. CoRR, arXiv:1503.04486 (2015)

  4. 4.

    Bienstock, D.: Some provably hard crossing number problems. Discret. Comput. Geom. 6(5), 443–459 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bienstock, D., Dean, N.: Bounds for rectilinear crossing numbers. J Graph Theory 17(3), 333–348 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and real computation. Springer-Verlag, New York (1998)

    Google Scholar 

  7. 7.

    Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. (N.S.) 21(1), 1–46 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Buss, J.F., Frandsen, G.S., Shallit, J.O.: The computational complexity of some problems of linear algebra. J. Comput. System Sci. 58(3), 572–596 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Canny, J.: Some algebraic and geometric computations in pspace. In: STOC ’88: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 460–469. ACM, NY, USA (1988)

  10. 10.

    Davis, M., Matijasevič, Y., Robinson, J.: Hilbert’s tenth problem: Diophantine equations: positive aspects of a negative solution. In: Mathematical Developments Arising from Hilbert Problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), pp. 323–378. (loose erratum). American Mathematics Society, Providence, RI (1976)

  11. 11.

    Håstad, J.: Tensor rank is NP-complete. In: Automata, languages and programming (Stresa, 1989), volume 372 of Lecture Notes in Computer Science, pp. 451–460. Springer, Berlin (1989)

  12. 12.

    Håstad, J.: Tensor rank is NP-complete. J. Algorithm. 11(4), 644–654 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Hillar, C.J., Lim, L.-H.: Most tensor problems are NP-hard. J. ACM 60(6), Art. 45, 39 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Howell, T.D.: Global properties of tensor rank. Linear Algebra Appl. 22, 9–23 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Koenigsmann, J.: Defining \(\mathbb {Z}\) in \(\mathbb {Q}\) ArXiv e-prints (2010)

  16. 16.

    Koiran, P.: Hilbert’s Nullstellensatz is in the polynomial hierarchy. J. Complex. 12(4), 273–286 (1996). Special issue for the Foundations of Computational Mathematics Conference (Rio de Janeiro, 1997)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

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

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Combin. Theory Ser. B 62(2), 289–315 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Matijasevič, J.V.: The Diophantineness of enumerable sets. Dokl. Akad. Nauk SSSR 191, 279–282 (1970)

    MathSciNet  Google Scholar 

  20. 20.

    Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Topology and geometry—Rohlin Seminar, volume 1346 of Lecture Notes in Mathematics, pp. 527–543. Springer, Berlin (1988)

  21. 21.

    Poonen, B.: Characterizing integers among rational numbers with a universal-existential formula. Amer. J. Math. 131(3), 675–682 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Richter-Gebert, J.: Mnëv’s universality theorem revisited. Sém Lothar. Combin., pp. 34 (1995)

  23. 23.

    Richter-Gebert, J.: Realization spaces of polytopes, volume 1643 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1996)

    Google Scholar 

  24. 24.

    Schaefer, M.: Complexity of some geometric and topological problems. In: Eppstein, D., Gansner, E.R. (eds.) Graph Drawing, volume 5849 of Lecture Notes in Computer Science, pp. 334–344. Springer (2009)

  25. 25.

    Schaefer, M.: Realizability of graphs and linkages. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 461–482. Springer (2012)

  26. 26.

    Schaefer, M., Štefankovič, D.: Fixed points Nash equilibria, and the existential theory of the reals. Theory of Computing Systems, pp. 1–22 (2015)

  27. 27.

    Shitov, Y.: How hard is the tensor rank?. CoRR, arXiv:1611.01559 (2016)

  28. 28.

    Shor, P.W.: Stretchability of pseudolines is NP-hard. In: Applied geometry and discrete mathematics, volume 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pp. 531–554. American Mathematics Society, Providence, RI (1991)

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Schaefer, M., Štefankovič, D. The Complexity of Tensor Rank. Theory Comput Syst 62, 1161–1174 (2018). https://doi.org/10.1007/s00224-017-9800-y

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Keywords

  • Computational complexity
  • Existential theory of fields
  • Tensor rank