A low-rank approximation of tensors and the topo-logical group structure of invertible matrices

R.N. Gumerov, A.S. Sharafutdinov

Kazan Federal University, Kazan, 420008 Russia

Full text PDF

For citation: Gumerov R.N., Sharafutdinov A.S. A low-rank approximation of tensors and the topological group structure of invertible matrices. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 4, pp. 788–796.

Для цитирования: Gumerov R.N., Sharafutdinov A.S. A low-rank approximation of tensors and the topological group structure of invertible matrices // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. – 2018. – Т. 160, кн. 4. – С. 788–796. 

Abstract

By a tensor we mean an element of the tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, i.e., represented as an array consisting of numbers. The properties of the tensor rank, which is a natural generalization of the matrix rank, have been considered in this paper. The topological group structure of invertible matrices has been studied. The multilinear matrix multiplication has been discussed from the viewpoint of transformation groups. We treat a low-rank tensor approximation in finite-dimensional tensor products. It has been shown that the problem on determining the best rank-n approximation for a tensor of size n×n×2 has no solution. To this end, we have used an approximation by matrices with simple spectra.

Keywords: approximation by matrices with simple spectra, group action, low-rank tensor approximation, norm on tensor space, open mapping, simple spectrum of matrix, tensor rank, topological group of invertible matrices, topological transformation group

Acknowledgments. The research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project no. 1.13556.2019/13.1.

The authors are grateful to Professors M.M. Karchevskii, R.R. Shagidullin and the participants of the seminar “Tensor Analysis” held at Kazan Federal University for their helpful discussions of the problems covered in this paper. We thank Professors G.G. Amosov, M. Fragoulopoulou, and A.Ya. Helemskii for their discussions during the Conference “Probability Theory and Mathematical Statistics” (November 7–10, 2017, Kazan).

References

  1. Helemskii A.Ya. Quantum functional analysis: Non-coordinate approach. In: University Lecture Series. Providence, Rhode Island, Amer. Math. Soc., 2010, vol. 56. 241 p.
  2. Hackbusch W. Tensor Spaces and Numerical Tensor Calculus. Berlin, Springer, 2012. xxiv, 500 p. doi: 10.1007/978-3-642-28027-6.
  3. Landsberg J.M. Tensors: Geometry and applications. In: Graduate Studies in Mathematics. Providence, Rhode Island, Amer. Math. Soc, 2012, vol. 128. 439 p.
  4. de Silva V., Lim L.-H. Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl., 2008, vol. 30, no. 3, pp. 1084–1127. doi: 10.1137/06066518X.
  5. Bini D., Capovani M., Lotti G., Romani F. O(n2.7799) complexity for n×n approximate matrix multiplication. Inf. Process. Lett., 1979, vol. 8, pp. 234–235.
  6. Gumerov R.N. Homological dimension of radical algebras of Beurling type with a rapidly decreasing weight. Moscow Univ. Math. Bull., 1988, vol. 43, no. 5, pp. 24–28.
  7. Gumerov R.N. On a geometric test for nontriviality of the simplicial and cyclic cohomology of Banach algebras. Moscow Univ. Math. Bull., 1991, vol. 46, no. 3, pp. 52–53.
  8. Tyrtyshnikov E.E. Tensor ranks for the inversion of tensor-product binomials. J. Comput. Appl. Math., 2010, vol. 234, no. 11, pp. 3170–3174. doi: 10.1016/j.cam.2010.02.006.
  9. Gumerov R.N., Vidunov S.I. Approximation by matrices with simple spectra. Lobachevskii J. Math., 2016, vol. 37, no. 3, pp. 240–243. doi: 10.1134/S1995080216030112.
  10. Greub W.H. Multilinear Algebra. Berlin, Heidelberg, Springer-Verlag, 1967. 225 p. doi: 10.1007/978-3-662-00795-2.
  11. Helemskii A.Ya. Lectures and exercises on functional analysis. In: Translations of Mathematical Monographs. Providence, Amer. Math. Soc., 2006, vol. 233. 470 p.
  12. Bredon G.E. Introduction to Compact Transformation Groups. New York, Acad. Press, 1972. 461 p.
  13. Vries de J. Elements of Topological Dynamics. Dordrecht, Springer, 1993. xvi, 748 p. doi: 10.1007/978-94-015-8171-4.
  14. Sakata T., Sumi T., Miyazaki M. Algebraic and Computational Aspects of Real Tensor Ranks. Tokyo, Springer, 2016. viii, 108 p. doi: 10.1007/978-4-431-55459-2.
  15. Hewitt E., Ross K.A. Abstract Harmonic Analysis. Vol. 1. Berlin, Springer, 1963. 519 p. doi: 10.1007/978-1-4419-8638-2.
  16. Ja’Ja’ J. Optimal evaluation of a pair of bilinear forms. SIAM J. Comput., 1979, vol. 8, no. 3, pp. 281–293. doi: 10.1137/0208037.
  17. Sumi T., Miyazaki M., Sakata T. Rank of 3-tensors with 2 slices and Kronecker canonical forms. Linear Algebra Appl., 2009, vol. 431, no. 10, pp. 1858–1868. doi: 10.1016/j.laa.2009.06.023.

Received

October 25, 2017

 

Gumerov Renat Nelsonovich, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematical Analysis

Kazan Federal University

ul. Kremlevskaya, 18, Kazan, 420008 Russia

E-mail: Renat.Gumerov@kpfu.ru

 

Sharafutdinov Azat Shamilevich, Student of Lobachevsky Institute of Mathematics and Mechanics

Kazan Federal University

ul. Kremlevskaya, 18, Kazan, 420008 Russia

E-mail: shash1996@mail.ru

 

Контент доступен под лицензией Creative Commons Attribution 4.0 License.