Worst-case results for positive semidefinite rank
Autor(a) principal: | |
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Data de Publicação: | 2015 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10316/44195 https://doi.org/10.1007/s10107-015-0867-4 |
Resumo: | We present various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv)^{\frac{1}{4}} improving on previous lower bounds. For polygons with v vertices, we show that psd rank cannot exceed 4⌈v/6⌉ which in turn shows that the psd rank of a p×q matrix of rank three is at most 4⌈min{p,q}/6⌉. In general, a nonnegative matrix of rank {k+1 \atopwithdelims ()2} has psd rank at least k and we pose the problem of deciding whether the psd rank is exactly k. Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when k is fixed. |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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7160 |
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Worst-case results for positive semidefinite rankWe present various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv)^{\frac{1}{4}} improving on previous lower bounds. For polygons with v vertices, we show that psd rank cannot exceed 4⌈v/6⌉ which in turn shows that the psd rank of a p×q matrix of rank three is at most 4⌈min{p,q}/6⌉. In general, a nonnegative matrix of rank {k+1 \atopwithdelims ()2} has psd rank at least k and we pose the problem of deciding whether the psd rank is exactly k. Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when k is fixed.Springer2015info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://hdl.handle.net/10316/44195http://hdl.handle.net/10316/44195https://doi.org/10.1007/s10107-015-0867-4https://doi.org/10.1007/s10107-015-0867-4enghttps://doi.org/10.1007/s10107-015-0867-4Gouveia, JoãoRobinson, Richard Z.Thomas, Rekha R.info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2021-06-29T10:02:52Zoai:estudogeral.uc.pt:10316/44195Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T20:53:31.903940Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Worst-case results for positive semidefinite rank |
title |
Worst-case results for positive semidefinite rank |
spellingShingle |
Worst-case results for positive semidefinite rank Gouveia, João |
title_short |
Worst-case results for positive semidefinite rank |
title_full |
Worst-case results for positive semidefinite rank |
title_fullStr |
Worst-case results for positive semidefinite rank |
title_full_unstemmed |
Worst-case results for positive semidefinite rank |
title_sort |
Worst-case results for positive semidefinite rank |
author |
Gouveia, João |
author_facet |
Gouveia, João Robinson, Richard Z. Thomas, Rekha R. |
author_role |
author |
author2 |
Robinson, Richard Z. Thomas, Rekha R. |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Gouveia, João Robinson, Richard Z. Thomas, Rekha R. |
description |
We present various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv)^{\frac{1}{4}} improving on previous lower bounds. For polygons with v vertices, we show that psd rank cannot exceed 4⌈v/6⌉ which in turn shows that the psd rank of a p×q matrix of rank three is at most 4⌈min{p,q}/6⌉. In general, a nonnegative matrix of rank {k+1 \atopwithdelims ()2} has psd rank at least k and we pose the problem of deciding whether the psd rank is exactly k. Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when k is fixed. |
publishDate |
2015 |
dc.date.none.fl_str_mv |
2015 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10316/44195 http://hdl.handle.net/10316/44195 https://doi.org/10.1007/s10107-015-0867-4 https://doi.org/10.1007/s10107-015-0867-4 |
url |
http://hdl.handle.net/10316/44195 https://doi.org/10.1007/s10107-015-0867-4 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
https://doi.org/10.1007/s10107-015-0867-4 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Springer |
publisher.none.fl_str_mv |
Springer |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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