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On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness

Authors Joshua A. Grochow , Youming Qiao

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ACM Subject Classification
  • Theory of computation → Complexity classes
  • Computing methodologies → Linear algebra algorithms
  • Hardware → Quantum communication and cryptography
Keywords
  • complexity class
  • tensor isomorphism
  • polynomial isomorphism
  • group isomorphism
  • stochastic local operations and classical communication

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Abstract

We study the complexity of isomorphism problems for tensors, groups, and polynomials. These problems have been studied in multivariate cryptography, machine learning, quantum information, and computational group theory. We show that these problems are all polynomial-time equivalent, creating bridges between problems traditionally studied in myriad research areas. This prompts us to define the complexity class TI, namely problems that reduce to the Tensor Isomorphism (TI) problem in polynomial time. Our main technical result is a polynomial-time reduction from d-tensor isomorphism to 3-tensor isomorphism. In the context of quantum information, this result gives multipartite-to-tripartite entanglement transformation procedure, that preserves equivalence under stochastic local operations and classical communication (SLOCC).

Cite As Get BibTex

Joshua A. Grochow and Youming Qiao. On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ITCS.2021.31

Author Details

Joshua A. Grochow
  • Departments of Computer Science and Mathematics, University of Colorado, Boulder, CO, USA
Youming Qiao
  • Centre for Quantum Software and Information, University of Technology Sydney, Australia

Funding

  • Grochow, Joshua A.: Partially supported by NSF grants DMS-1750319 and DMS-1622390.
  • Qiao, Youming: Partially supported by the Australian Research Council DP200100950.

Acknowledgements

We would like to thanks James B. Wilson for related discussions, and Uriya First, Lek-Heng Lim, and J. M. Landsberg for help searching for references asking whether dTI could reduce to 3TI. We also thank Nengkun Yu, Yinan Li, and Graeme Smith for explaining the notion of SLOCC, and Ryan Williams for pointing out the problem of distinguishing between ETH and #ETH.

References

  1. Manindra Agrawal and Nitin Saxena. Automorphisms of finite rings and applications to complexity of problems. In STACS 2005, 22nd Annual Symposium on Theoretical Aspects of Computer Science, Proceedings, pages 1-17, 2005. URL: https://doi.org/10.1007/978-3-540-31856-9_1.
  2. Manindra Agrawal and Nitin Saxena. Equivalence of 𝔽-algebras and cubic forms. In STACS 2006, 23rd Annual Symposium on Theoretical Aspects of Computer Science, Proceedings, pages 115-126, 2006. URL: https://doi.org/10.1007/11672142_8.
  3. Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. Inf. Comput., 256:2-8, 2017. URL: https://doi.org/10.1016/j.ic.2017.04.004.
  4. László Babai. On the automorphism groups of strongly regular graphs I. In Proceedings of the 5th Conference on Innovations in Theoretical Computer Science, ITCS '14, pages 359-368, 2014. URL: https://doi.org/10.1145/2554797.2554830.
  5. László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, pages 684-697, 2016. https://arxiv.org/abs/1512.03547 [cs.DS] version 2. URL: https://doi.org/10.1145/2897518.2897542.
  6. László Babai, Robert Beals, and Ákos Seress. Polynomial-time theory of matrix groups. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pages 55-64, 2009. URL: https://doi.org/10.1145/1536414.1536425.
  7. László Babai, Paolo Codenotti, Joshua A. Grochow, and Youming Qiao. Code equivalence and group isomorphism. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA11), pages 1395-1408, Philadelphia, PA, 2011. SIAM. URL: https://doi.org/10.1137/1.9781611973082.107.
  8. László Babai, Paul Erdős, and Stanley M. Selkow. Random graph isomorphism. SIAM J. Comput., 9(3):628-635, 1980. URL: https://doi.org/10.1137/0209047.
  9. Reinhold Baer. Groups with abelian central quotient group. Trans. AMS, 44(3):357-386, 1938. URL: https://doi.org/10.1090/S0002-9947-1938-1501972-1.
  10. Charles H. Bennett, Sandu Popescu, Daniel Rohrlich, John A. Smolin, and Ashish V. Thapliyal. Exact and asymptotic measures of multipartite pure-state entanglement. Physical Review A, 63(1):012307, 2000. URL: https://doi.org/10.1103/PhysRevA.63.012307.
  11. Jérémy Berthomieu, Jean-Charles Faugère, and Ludovic Perret. Polynomial-time algorithms for quadratic isomorphism of polynomials: The regular case. J. Complexity, 31(4):590-616, 2015. URL: https://doi.org/10.1016/j.jco.2015.04.001.
  12. W. Bosma, J. J. Cannon, and C. Playoust. The Magma algebra system I: the user language. J. Symb. Comput., pages 235-265, 1997. URL: https://doi.org/10.1006/jsco.1996.0125.
  13. Peter Brooksbank, E O'Brien, and James Wilson. Testing isomorphism of graded algebras. Trans. Amer. Math. Soc., 372:8067-8090, 2019. URL: https://doi.org/10.1090/tran/7884.
  14. Peter A. Brooksbank, Joshua A. Grochow, Yinan Li, Youming Qiao, and James B. Wilson. Incorporating Weisfeiler-Leman into algorithms for group isomorphism. https://arxiv.org/abs/1905.02518 [cs.CC], 2019. Google Scholar
  15. Peter A. Brooksbank and Eugene M. Luks. Testing isomorphism of modules. J. Algebra, 320(11):4020-4029, 2008. URL: https://doi.org/10.1016/j.jalgebra.2008.07.014.
  16. Peter A. Brooksbank, Joshua Maglione, and James B. Wilson. Thetensor.space. https://github.com/thetensor-space/, 2019.
  17. Peter A. Brooksbank and James B. Wilson. Computing isometry groups of Hermitian maps. Trans. Amer. Math. Soc., 364:1975-1996, 2012. URL: https://doi.org/10.1090/S0002-9947-2011-05388-2.
  18. Peter A Brooksbank and James B Wilson. The module isomorphism problem reconsidered. Journal of Algebra, 421:541-559, 2015. URL: https://doi.org/10.1016/j.jalgebra.2014.09.004.
  19. John Cannon and Derek F. Holt. Automorphism group computation and isomorphism testing in finite groups. J. Symbolic Comput., 35(3):241-267, 2003. URL: https://doi.org/10.1016/S0747-7171(02)00133-5.
  20. Kuo-Tsai Chen. Integration of paths, geometric invariants and a generalized baker-hausdorff formula. Annals of Mathematics, pages 163-178, 1957. URL: https://doi.org/10.2307/1969671.
  21. Ilya Chevyrev and Andrey Kormilitzin. A primer on the signature method in machine learning, 2016. URL: http://arxiv.org/abs/1603.03788.
  22. Alexander Chistov, Gábor Ivanyos, and Marek Karpinski. Polynomial time algorithms for modules over finite dimensional algebras. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ISSAC '97, pages 68-74. ACM, 1997. URL: https://doi.org/10.1145/258726.258751.
  23. Wolfgang Dür, Guifre Vidal, and J. Ignacio Cirac. Three qubits can be entangled in two inequivalent ways. Physical Review A, 62(6):062314, 2000. URL: https://doi.org/10.1103/PhysRevA.62.062314.
  24. V. Felsch and J. Neubüser. On a programme for the determination of the automorphism group of a finite group. In Pergamon J. Leech, editor, Computational Problems in Abstract Algebra (Proceedings of a Conference on Computational Problems in Algebra, Oxford, 1967), pages 59-60, Oxford, 1970. Google Scholar
  25. Lance Fortnow and Joshua A. Grochow. Complexity classes of equivalence problems revisited. Inform. and Comput., 209(4):748-763, 2011. Also available as https://arxiv.org/abs/0907.4775 [cs.CC]. URL: https://doi.org/10.1016/j.ic.2011.01.006.
  26. Vyacheslav Futorny, Joshua A. Grochow, and Vladimir V. Sergeichuk. Wildness for tensors. Lin. Alg. Appl., 566:212-244, 2019. URL: https://doi.org/10.1016/j.laa.2018.12.022.
  27. I. M. Gelfand and V. A. Ponomarev. Remarks on the classification of a pair of commuting linear transformations in a finite-dimensional space. Functional Anal. Appl., 3:325-326, 1969. URL: https://doi.org/10.1007/BF01076321.
  28. Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at URL: https://faculty.math.illinois.edu/Macaulay2/.
  29. D. Ju. Grigoriev. Complexity of "wild" matrix problems and of the isomorphism of algebras and graphs. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 105:10-17, 198, 1981. Theoretical applications of the methods of mathematical logic, III. URL: https://doi.org/10.1007/BF01084390.
  30. Joshua A. Grochow. Matrix Lie algebra isomorphism. In IEEE Conference on Computational Complexity (CCC12), pages 203-213, 2012. Also available as https://arxiv.org/abs/1112.2012 [cs.CC] and ECCC Technical Report TR11-168. URL: https://doi.org/10.1109/CCC.2012.34.
  31. Joshua A. Grochow and Youming Qiao. Algorithms for group isomorphism via group extensions and cohomology. SIAM J. Comput., 46(4):1153-1216, 2017. Preliminary version in IEEE Conference on Computational Complexity (CCC) 2014 (DOI:10.1109/CCC.2014.19). Also available as https://arxiv.org/abs/1309.1776 [cs.DS] and ECCC Technical Report TR13-123. URL: https://doi.org/10.1137/15M1009767.
  32. Joshua A. Grochow and Youming Qiao. Isomorphism problems for tensors, groups, and cubic forms: completeness and reductions. CoRR, abs/1907.00309, 2019. URL: http://arxiv.org/abs/1907.00309.
  33. Joshua A. Grochow and Youming Qiao. On isomorphism problems for tensors, groups, and polynomials II: search and counting to decision reductions, and applications to group isomorphism, 2020. Under preparation. Google Scholar
  34. Xiaoyu He and Youming Qiao. On the Baer-Lovász-Tutte construction of groups from graphs: isomorphism types and homomorphism notions, 2020. https://arxiv.org/abs/2003.07200 [math.CO]. Google Scholar
  35. Harald Andrés Helfgott, Jitendra Bajpai, and Daniele Dona. Graph isomorphisms in quasi-polynomial time. https://arxiv.org/abs/1710.04574 [math.GR], 2017. Google Scholar
  36. Graham Higman. Enumerating p-groups. I. Inequalities. Proc. London Math. Soc. (3), 10:24-30, 1960. URL: https://doi.org/10.1112/plms/s3-10.1.24.
  37. Gábor Ivanyos, Marek Karpinski, and Nitin Saxena. Deterministic polynomial time algorithms for matrix completion problems. SIAM J. Comput., 39(8):3736-3751, 2010. URL: https://doi.org/10.1137/090781231.
  38. Gábor Ivanyos and Youming Qiao. Algorithms based on *-algebras, and their applications to isomorphism of polynomials with one secret, group isomorphism, and polynomial identity testing. SIAM J. Comput., 48(3):926-963, 2019. URL: https://doi.org/10.1137/18M1165682.
  39. Zhengfeng Ji, Youming Qiao, Fang Song, and Aaram Yun. General linear group action on tensors: A candidate for post-quantum cryptography. In Dennis Hofheinz and Alon Rosen, editors, Theory of Cryptography - 17th International Conference, TCC 2019, Nuremberg, Germany, December 1-5, 2019, Proceedings, Part I, volume 11891 of Lecture Notes in Computer Science, pages 251-281. Springer, 2019. Preprint https://arxiv.org/abs/1906.04330 [cs.CR]. URL: https://doi.org/10.1007/978-3-030-36030-6_11.
  40. Neeraj Kayal. Efficient algorithms for some special cases of the polynomial equivalence problem. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 1409-1421, 2011. URL: https://doi.org/10.1137/1.9781611973082.108.
  41. Neeraj Kayal. Affine projections of polynomials: extended abstract. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 643-662, 2012. URL: https://doi.org/10.1145/2213977.2214036.
  42. Neeraj Kayal and Nitin Saxena. Complexity of ring morphism problems. Computational Complexity, 15(4):342-390, 2006. URL: https://doi.org/10.1007/s00037-007-0219-8.
  43. Johannes Köbler, Uwe Schöning, and Jacobo Torán. The graph isomorphism problem: its structural complexity. Birkhauser Verlag, Basel, Switzerland, Switzerland, 1993. URL: https://doi.org/10.1007/978-1-4612-0333-9.
  44. Pascal Koiran. Hilbert’s Nullstellensatz is in the polynomial hierarchy. J. Complexity, 12(4):273-286, 1996. URL: https://doi.org/10.1006/jcom.1996.0019.
  45. J.M. Landsberg. Tensors: Geometry and Applications, volume 128 of Graduate studies in mathematics. American Mathematical Soc., 2012. URL: https://doi.org/10.1090/gsm/128.
  46. Yinan Li and Youming Qiao. Linear algebraic analogues of the graph isomorphism problem and the Erdős-Rényi model. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, pages 463-474. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.49.
  47. Eugene M. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci., 25(1):42-65, 1982. URL: https://doi.org/10.1016/0022-0000(82)90009-5.
  48. Eugene M. Luks. Permutation groups and polynomial-time computation. In Groups and computation (New Brunswick, NJ, 1991), volume 11 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 139-175. Amer. Math. Soc., Providence, RI, 1993. Google Scholar
  49. T. J. Lyons. Rough paths, signatures and the modelling of functions on streams. In Proc. International Congress of Mathematicians, pages 163-184. Kyung Moon Publishers, 2014. Google Scholar
  50. Terry J. Lyons and Weijun Xu. Inverting the signature of a path. J. Eur. Math. Soc.(JEMS), 20(7):1655-1687, 2018. URL: https://doi.org/10.4171/JEMS/796.
  51. Brendan D. McKay. Practical graph isomorphism. Congr. Numer., pages 45-87, 1980. Google Scholar
  52. Brendan D. McKay and Adolfo Piperno. Practical graph isomorphism, II. Journal of Symbolic Computation, 60(0):94-112, 2014. URL: https://doi.org/10.1016/j.jsc.2013.09.003.
  53. Gary L. Miller. On the n^log n isomorphism technique (a preliminary report). In STOC, pages 51-58. ACM, 1978. URL: https://doi.org/10.1145/800133.804331.
  54. P. J. Moore, T. J. Lyons, and J. Gallacher. Using path signatures to predict a diagnosis of alzheimer’s disease. PLoS ONE, 14(9), 2019. URL: https://doi.org/10.1371/journal.pone.0222212.
  55. Eamonn A O'Brien. Isomorphism testing for p-groups. Journal of Symbolic Computation, 17(2):133-147, 1994. URL: https://doi.org/10.1006/jsco.1994.1007.
  56. Rufus Oldenburger. Non-singular multilinear forms and certain p-way matrix factorizations. Trans. Amer. Math. Soc., 39(3):422-455, 1936. URL: https://doi.org/10.2307/1989760.
  57. Jacques Patarin. Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms. In Advances in Cryptology - EUROCRYPT '96, International Conference on the Theory and Application of Cryptographic Techniques, Saragossa, Spain, May 12-16, 1996, Proceeding, pages 33-48, 1996. URL: https://doi.org/10.1007/3-540-68339-9_4.
  58. Erez Petrank and Ron M. Roth. Is code equivalence easy to decide? IEEE Trans. Inf. Theory, 43(5):1602-1604, 1997. URL: https://doi.org/10.1109/18.623157.
  59. Max Pfeffer, Anna Seigal, and Bernd Sturmfels. Learning paths from signature tensors. SIAM Journal on Matrix Analysis and Applications, 40(2):394-416, 2019. arXiv:1809.01588. URL: https://doi.org/10.1137/18M1212331.
  60. Bjorn Poonen. Undecidable problems: a sampler. In Interpreting Gödel, pages 211-241. Cambridge Univ. Press, Cambridge, 2014. https://arxiv.org/abs/1204.0299 [math.LO]. Google Scholar
  61. Lajos Rónyai. Zero divisors in quaternion algebras. J. Algorithms, 9(4):494-506, 1988. URL: https://doi.org/10.1016/0196-6774(88)90014-4.
  62. David J. Rosenbaum. Bidirectional collision detection and faster deterministic isomorphism testing. arXiv preprint https://arxiv.org/abs/1304.3935 [cs.DS], 2013. Google Scholar
  63. David J. Rosenbaum. Breaking the n^log n barrier for solvable-group isomorphism. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1054-1073. SIAM, 2013. Preprint https://arxiv.org/abs/1205.0642 [cs.DS]. Google Scholar
  64. Nitin Saxena. Morphisms of rings and applications to complexity. PhD thesis, Indian Institute of Technology, Kanpur, May 2006. URL: https://www.cse.iitk.ac.in/users/nitin/papers/thesis.pdf.
  65. Ákos Seress. Permutation group algorithms, volume 152. Cambridge University Press, 2003. URL: https://doi.org/10.1017/CBO9780511546549.
  66. V. V. Sergeichuk. The classification of metabelian p-groups. In Matrix problems (Russian), pages 150-161. Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 1977. Google Scholar
  67. Vladimir V. Sergeichuk. Canonical matrices for linear matrix problems. Linear Algebra Appl., 317(1-3):53-102, 2000. URL: https://doi.org/10.1016/S0024-3795(00)00150-6.
  68. Charles C Sims. Some group-theoretic algorithms. In Topics in algebra, pages 108-124. Springer, 1978. URL: https://doi.org/10.1007/BFb0103126.
  69. James B. Wilson. Decomposing p-groups via Jordan algebras. J. Algebra, 322:2642-2679, 2009. URL: https://doi.org/10.1016/j.jalgebra.2009.07.029.
  70. James B. Wilson. 2014 conference on Groups, Computation, and Geometry at Colorado State University, co-organized by P. Brooksbank, A. Hulpke, T. Penttila, J. Wilson, and W. Kantor. Personal communication, 2014. Google Scholar
  71. James B. Wilson. Surviving in the wilderness. Talk presented at the Sante Fe Institute Workshop on Wildness in Computer Science, Physics, and Mathematics, 2015. Google Scholar
  72. SM Zangi, Jun-Li Li, and Cong-Feng Qiao. Quantum state concentration and classification of multipartite entanglement. Physical Review A, 97(1):012301, 2018. URL: https://doi.org/10.1103/PhysRevA.97.012301.
  73. V. N. Zemlyachenko, N. M. Korneenko, and R. I. Tyshkevich. Graph isomorphism problem. J. Soviet Math., 29(4):1426-1481, May 1985. URL: https://doi.org/10.1007/BF02104746.
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