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The span of singular tuples of a tensor beyond the boundary format
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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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en
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19
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Journal of Symbolic Computation, Volume 120, pp. 1-19
Abstract
A singular k-tuple of a tensor T of format (n1,…,nk) is essentially a complex critical point of the distance function from T constrained to the cone of tensors of format (n1,…,nk) of rank at most one. A generic tensor has finitely many complex singular k-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first k−1 dimensions ni, then the number of singular k-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable nk, that stabilizes when (n1,…,nk) reaches a boundary format. In this paper, we study the linear span of singular k-tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order-k tensors of format (2,…,2,n). As a consequence, if again we fix the first k−1 dimensions ni and let nk increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with k>3. Finally, we provide equations for the linear span of singular triples of a generic order three tensor T of some special non-sub-boundary format. From these equations, we conclude that T belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.
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Funding Information: We thank Giorgio Ottaviani for encouraging this project and for valuable guidance. A special acknowledgment goes to Paul Breiding and Sascha Timme for their support in coding the algorithm of Section 7 . The first author is partially supported by the Academy of Finland Grant 323416 . The second author is supported by European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Actions , grant agreement 813211 (POEMA). Publisher Copyright: © 2023 The Author(s)
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Sodomaco, L & Teixeira Turatti, E 2024, 'The span of singular tuples of a tensor beyond the boundary format', Journal of Symbolic Computation, vol. 120, 102230, pp. 1-19. https://doi.org/10.1016/j.jsc.2023.102230