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

R.N. Gumerov*, A.S. Sharafutdinov**

Kazan Federal University, Kazan, 420008 Russia

E-mail: *Renat.Gumerov@kpfu.ru, **shash1996@mail.ru

Received October 25, 2017

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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).

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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. 

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