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.
Similar content being viewed by others
Notes
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].
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.
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 \(\emptyset ^{\prime }\).
In other models, e.g., the Blum-Shub-Smale model [7] this was well-known earlier.
There are other definitions of eigenvalues for tensors as well.
The proof in [25] yields a quartic systems, but that can be reduced to quadratic, by removing the final (unnecessary) squaring operation.
References
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)
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)
Bhangale, A., Kopparty, S.: The complexity of computing the minimum rank of a sign pattern matrix. CoRR, arXiv:1503.04486 (2015)
Bienstock, D.: Some provably hard crossing number problems. Discret. Comput. Geom. 6(5), 443–459 (1991)
Bienstock, D., Dean, N.: Bounds for rectilinear crossing numbers. J Graph Theory 17(3), 333–348 (1993)
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and real computation. Springer-Verlag, New York (1998)
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)
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)
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)
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)
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)
Håstad, J.: Tensor rank is NP-complete. J. Algorithm. 11(4), 644–654 (1990)
Hillar, C.J., Lim, L.-H.: Most tensor problems are NP-hard. J. ACM 60(6), Art. 45, 39 (2013)
Howell, T.D.: Global properties of tensor rank. Linear Algebra Appl. 22, 9–23 (1978)
Koenigsmann, J.: Defining \(\mathbb {Z}\) in \(\mathbb {Q}\) ArXiv e-prints (2010)
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)
Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)
Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Combin. Theory Ser. B 62(2), 289–315 (1994)
Matijasevič, J.V.: The Diophantineness of enumerable sets. Dokl. Akad. Nauk SSSR 191, 279–282 (1970)
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)
Poonen, B.: Characterizing integers among rational numbers with a universal-existential formula. Amer. J. Math. 131(3), 675–682 (2009)
Richter-Gebert, J.: Mnëv’s universality theorem revisited. Sém Lothar. Combin., pp. 34 (1995)
Richter-Gebert, J.: Realization spaces of polytopes, volume 1643 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1996)
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)
Schaefer, M.: Realizability of graphs and linkages. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 461–482. Springer (2012)
Schaefer, M., Štefankovič, D.: Fixed points Nash equilibria, and the existential theory of the reals. Theory of Computing Systems, pp. 1–22 (2015)
Shitov, Y.: How hard is the tensor rank?. CoRR, arXiv:1611.01559 (2016)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-017-9800-y