It is well known that tensor decompositions show separations, that is, constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here, we show that many of these separations disappear in the approximate case. Specifically, for every approximation error ɛ and norm, we define the approximate rank as the minimum rank of an element in the ɛ-ball with respect to that norm. For positive semidefinite matrices, we show that the separations between rank, purification rank, and separable rank disappear for a large class of Schatten p-norms. For non-negative tensors, we show that the separations between rank, positive semidefinite rank, and non-negative rank disappear for all ℓp-norms with p > 1. For the trace norm (p = 1), we obtain upper bounds that depend on the ambient dimension. We also provide a deterministic algorithm to obtain the approximate decomposition attaining our bounds. Our main tool is an approximate version of the Carathéodory theorem. Our results imply that many separations are not robust under small perturbations of the tensor, with implications in quantum many-body systems and communication complexity.
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September 2021
Research Article|
September 01 2021
Approximate tensor decompositions: Disappearance of many separations
Gemma De las Cuevas
;
Gemma De las Cuevas
1
Institute for Theoretical Physics
, Technikerstr. 21a, A-6020 Innsbruck, Austria
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Andreas Klingler
;
Andreas Klingler
a)
1
Institute for Theoretical Physics
, Technikerstr. 21a, A-6020 Innsbruck, Austria
a)Author to whom correspondence should be addressed: [email protected]
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Tim Netzer
Tim Netzer
a)
2
Department of Mathematics
, Technikerstr. 13, A-6020 Innsbruck, Austria
a)Author to whom correspondence should be addressed: [email protected]
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Gemma De las Cuevas
1
Andreas Klingler
1,a)
Tim Netzer
2,a)
1
Institute for Theoretical Physics
, Technikerstr. 21a, A-6020 Innsbruck, Austria
2
Department of Mathematics
, Technikerstr. 13, A-6020 Innsbruck, Austria
a)Author to whom correspondence should be addressed: [email protected]
J. Math. Phys. 62, 093502 (2021)
Article history
Received:
October 20 2020
Accepted:
August 05 2021
Citation
Gemma De las Cuevas, Andreas Klingler, Tim Netzer; Approximate tensor decompositions: Disappearance of many separations. J. Math. Phys. 1 September 2021; 62 (9): 093502. https://doi.org/10.1063/5.0033876
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