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Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems

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Bibliographic Details
Title: Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems
Authors: Blomenhofer, Alexander Taveira, Laurent, Monique
Publication Year: 2024
Collection: ArXiv.org (Cornell University Library)
Subject Terms: Optimization and Control, 90C23, 90C22, 81P45
Description: 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
Document Type: text
Language: unknown
Relation: http://arxiv.org/abs/2412.13191
Availability: http://arxiv.org/abs/2412.13191
Accession Number: edsbas.594F2AF7
Database: BASE
Description
Abstract: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