Tensor decompositions and rank increment conjecture
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Eugene E. Tyrtyshnikov
Abstract
Some properties of tensor ranks and the non-closeness issue of sets with given restrictions on the rank of tensors entering those sets are studied. It is proved that the rank of the d-dimensional Laplacian equals d. The following conjecture is formulated: for any tensor of non-maximal rank there exists a nonzero decomposable tensor (tensor of rank 1) such that the rank increases by one after adding this tensor. In the general case, it is proved that this property holds algebraically almost everywhere for complex tensors of fixed size whose rank is strictly less than the generic rank.
Acknowledgment
The author is grateful to Martin Mohlenkamp for discussion of the problem of the Laplacian and the proposed idea allowing us to obtain the proof.
Funding: The work was supported by the Russian Science Foundation (project No. 19–11–00338).
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© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Preface
- Methods of variational data assimilation with application to problems of hydrothermodynamics of marine water areas
- Mathematical immunology: from phenomenological to multiphysics modelling
- Iterative solution methods for elliptic boundary value problems
- Multi-physics flux coupling for hydraulic fracturing modelling within INMOST platform
- Tensor decompositions and rank increment conjecture
- Global optimization based on TT-decomposition
Artikel in diesem Heft
- Frontmatter
- Preface
- Methods of variational data assimilation with application to problems of hydrothermodynamics of marine water areas
- Mathematical immunology: from phenomenological to multiphysics modelling
- Iterative solution methods for elliptic boundary value problems
- Multi-physics flux coupling for hydraulic fracturing modelling within INMOST platform
- Tensor decompositions and rank increment conjecture
- Global optimization based on TT-decomposition
