The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations p...

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Detalles Bibliográficos
Autores: Beltrán Álvarez, Carlos|||0000-0002-0689-8232, Breiding, Paul, Vannieuwenhoven, Nick
Tipo de recurso: artículo
Fecha de publicación:2022
País:España
Institución:Universidad de Cantabria (UC)
Repositorio:UCrea Repositorio Abierto de la Universidad de Cantabria
Idioma:inglés
OAI Identifier:oai:repositorio.unican.es:10902/24539
Acceso en línea:http://hdl.handle.net/10902/24539
Access Level:acceso abierto
Palabra clave:Tensor decomposition
Condition number
Average analysis
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spelling The Average Condition Number of Most Tensor Rank Decomposition Problems is InfiniteBeltrán Álvarez, Carlos|||0000-0002-0689-8232Breiding, PaulVannieuwenhoven, NickTensor decompositionCondition numberAverage analysisThe tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks r?3r?3 as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions.We thank the reviewers for helpful suggestions. Part of this work was made while the second and third author were visiting the Universidad de Cantabria, supported by the funds of Grant 21.SI01.64658 (Banco Santander and Universidad de Cantabria), Grant MTM2017-83816-P from the Spanish Ministry of Science. The third author was additionally supported by the FWO Grant for a long stay abroad V401518N. We thank these institutions for their support.Springer New York LLCUniversidad de Cantabria20222022-02-01journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articlehttp://hdl.handle.net/10902/24539Foundations of Computational Mathematics, 2022reponame:UCrea Repositorio Abierto de la Universidad de Cantabriainstname:Universidad de Cantabria (UC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nd/4.0/info:eu-repo/semantics/openAccessoai:repositorio.unican.es:10902/245392026-06-02T12:39:31Z
dc.title.none.fl_str_mv The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite
title The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite
spellingShingle The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite
Beltrán Álvarez, Carlos|||0000-0002-0689-8232
Tensor decomposition
Condition number
Average analysis
title_short The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite
title_full The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite
title_fullStr The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite
title_full_unstemmed The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite
title_sort The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite
dc.creator.none.fl_str_mv Beltrán Álvarez, Carlos|||0000-0002-0689-8232
Breiding, Paul
Vannieuwenhoven, Nick
author Beltrán Álvarez, Carlos|||0000-0002-0689-8232
author_facet Beltrán Álvarez, Carlos|||0000-0002-0689-8232
Breiding, Paul
Vannieuwenhoven, Nick
author_role author
author2 Breiding, Paul
Vannieuwenhoven, Nick
author2_role author
author
dc.contributor.none.fl_str_mv Universidad de Cantabria
dc.subject.none.fl_str_mv Tensor decomposition
Condition number
Average analysis
topic Tensor decomposition
Condition number
Average analysis
description The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks r?3r?3 as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions.
publishDate 2022
dc.date.none.fl_str_mv 2022
2022-02-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10902/24539
url http://hdl.handle.net/10902/24539
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nd/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nd/4.0/
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Springer New York LLC
publisher.none.fl_str_mv Springer New York LLC
dc.source.none.fl_str_mv Foundations of Computational Mathematics, 2022
reponame:UCrea Repositorio Abierto de la Universidad de Cantabria
instname:Universidad de Cantabria (UC)
instname_str Universidad de Cantabria (UC)
reponame_str UCrea Repositorio Abierto de la Universidad de Cantabria
collection UCrea Repositorio Abierto de la Universidad de Cantabria
repository.name.fl_str_mv
repository.mail.fl_str_mv
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