A low-rank approximation of tensors and the topological group structure of invertible matrices
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 t...
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Kazan Federal University
2018-12-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| author | R.N. Gumerov A.S. Sharafutdinov |
| author_facet | R.N. Gumerov A.S. Sharafutdinov |
| author_sort | R.N. Gumerov |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-d4913c7bc3fa406cafba84c34ca5403d |
| institution | Kabale University |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2018-12-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-d4913c7bc3fa406cafba84c34ca5403d2025-01-03T00:04:31ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982018-12-011604788796A low-rank approximation of tensors and the topological group structure of invertible matricesR.N. Gumerov0A.S. Sharafutdinov1Kazan Federal University, Kazan, 420008 RussiaKazan Federal University, Kazan, 420008 RussiaBy 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.https://kpfu.ru/a-low-rank-approximation-of-tensors-and-the-403972.htmlapproximation by matrices with simple spectragroup actionlow-rank tensor approximationnorm on tensor spaceopen mappingsimple spectrum of matrixtensor ranktopological group of invertible matricestopological transformation group |
| spellingShingle | R.N. Gumerov A.S. Sharafutdinov A low-rank approximation of tensors and the topological group structure of invertible matrices Учёные записки Казанского университета: Серия Физико-математические науки 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 |
| title | A low-rank approximation of tensors and the topological group structure of invertible matrices |
| title_full | A low-rank approximation of tensors and the topological group structure of invertible matrices |
| title_fullStr | A low-rank approximation of tensors and the topological group structure of invertible matrices |
| title_full_unstemmed | A low-rank approximation of tensors and the topological group structure of invertible matrices |
| title_short | A low-rank approximation of tensors and the topological group structure of invertible matrices |
| title_sort | low rank approximation of tensors and the topological group structure of invertible matrices |
| topic | 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 |
| url | https://kpfu.ru/a-low-rank-approximation-of-tensors-and-the-403972.html |
| work_keys_str_mv | AT rngumerov alowrankapproximationoftensorsandthetopologicalgroupstructureofinvertiblematrices AT assharafutdinov alowrankapproximationoftensorsandthetopologicalgroupstructureofinvertiblematrices AT rngumerov lowrankapproximationoftensorsandthetopologicalgroupstructureofinvertiblematrices AT assharafutdinov lowrankapproximationoftensorsandthetopologicalgroupstructureofinvertiblematrices |