Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems
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| Titel: | Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems |
|---|---|
| Autoren: | Blomenhofer, Alexander Taveira, Laurent, Monique |
| Publikationsjahr: | 2024 |
| Bestand: | ArXiv.org (Cornell University Library) |
| Schlagwörter: | Optimization and Control, 90C23, 90C22, 81P45 |
| Beschreibung: | We revisit the convergence analysis of two approximation hierarchies for polynomial optimization on the unit sphere. The first one is based on the moment-sos approach and gives semidefinite bounds for which Fang and Fawzi (2021) showed an analysis in $O(1/r^2)$ for the r-th level bound, using the polynomial kernel method. The second hierarchy was recently proposed by Lovitz and Johnston (2023) and gives spectral bounds for which they show a convergence rate in $O(1/r)$, using a quantum de Finetti theorem of Christandl et al. (2007) that applies to complex Hermitian matrices with a "double" symmetry. We investigate links between these approaches, in particular, via duality of moments and sums of squares. Our main results include showing that the spectral bounds cannot have a convergence rate better than $O(1/r^2)$ and that they do not enjoy generic finite convergence. In addition, we propose alternative performance analyses that involve explicit constants depending on intrinsic parameters of the optimization problem. For this we develop a novel "banded" real de Finetti theorem that applies to real matrices with "double" symmetry. We also show how to use the polynomial kernel method to obtain a de Finetti type result in $O(1/r^2)$ for real maximally symmetric matrices, improving an earlier result in $O(1/r)$ of Doherty and Wehner (2012). ; 31 pages |
| Publikationsart: | text |
| Sprache: | unknown |
| Relation: | http://arxiv.org/abs/2412.13191 |
| Verfügbarkeit: | http://arxiv.org/abs/2412.13191 |
| Dokumentencode: | edsbas.594F2AF7 |
| Datenbank: | BASE |
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| Items | – Name: Title Label: Title Group: Ti Data: Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Blomenhofer%2C+Alexander+Taveira%22">Blomenhofer, Alexander Taveira</searchLink><br /><searchLink fieldCode="AR" term="%22Laurent%2C+Monique%22">Laurent, Monique</searchLink> – Name: DatePubCY Label: Publication Year Group: Date Data: 2024 – Name: Subset Label: Collection Group: HoldingsInfo Data: ArXiv.org (Cornell University Library) – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Optimization+and+Control%22">Optimization and Control</searchLink><br /><searchLink fieldCode="DE" term="%2290C23%22">90C23</searchLink><br /><searchLink fieldCode="DE" term="%2290C22%22">90C22</searchLink><br /><searchLink fieldCode="DE" term="%2281P45%22">81P45</searchLink> – Name: Abstract Label: Description Group: Ab Data: We revisit the convergence analysis of two approximation hierarchies for polynomial optimization on the unit sphere. The first one is based on the moment-sos approach and gives semidefinite bounds for which Fang and Fawzi (2021) showed an analysis in $O(1/r^2)$ for the r-th level bound, using the polynomial kernel method. The second hierarchy was recently proposed by Lovitz and Johnston (2023) and gives spectral bounds for which they show a convergence rate in $O(1/r)$, using a quantum de Finetti theorem of Christandl et al. (2007) that applies to complex Hermitian matrices with a "double" symmetry. We investigate links between these approaches, in particular, via duality of moments and sums of squares. Our main results include showing that the spectral bounds cannot have a convergence rate better than $O(1/r^2)$ and that they do not enjoy generic finite convergence. In addition, we propose alternative performance analyses that involve explicit constants depending on intrinsic parameters of the optimization problem. For this we develop a novel "banded" real de Finetti theorem that applies to real matrices with "double" symmetry. We also show how to use the polynomial kernel method to obtain a de Finetti type result in $O(1/r^2)$ for real maximally symmetric matrices, improving an earlier result in $O(1/r)$ of Doherty and Wehner (2012). ; 31 pages – Name: TypeDocument Label: Document Type Group: TypDoc Data: text – Name: Language Label: Language Group: Lang Data: unknown – Name: NoteTitleSource Label: Relation Group: SrcInfo Data: http://arxiv.org/abs/2412.13191 – Name: URL Label: Availability Group: URL Data: http://arxiv.org/abs/2412.13191 – Name: AN Label: Accession Number Group: ID Data: edsbas.594F2AF7 |
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| RecordInfo | BibRecord: BibEntity: Languages: – Text: unknown Subjects: – SubjectFull: Optimization and Control Type: general – SubjectFull: 90C23 Type: general – SubjectFull: 90C22 Type: general – SubjectFull: 81P45 Type: general Titles: – TitleFull: Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Blomenhofer, Alexander Taveira – PersonEntity: Name: NameFull: Laurent, Monique IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Type: published Y: 2024 Identifiers: – Type: issn-locals Value: edsbas – Type: issn-locals Value: edsbas.oa |
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