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Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems
Taveira Blomenhofer,Alexander Laurent,Monique
Taveira Blomenhofer,Alexander
Laurent,Monique
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/r2) for the rth 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/r2) 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/r2) for real maximally symmetric matrices, improving an earlier result in O(1/r) of Doherty and Wehner (2012).
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Date
2025-10
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2412.13191v3.pdf
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Keywords
polynomial optimization, sums of squares, quantum de Finetti, semi-definite programming, spectral bounds, moment problem
Citation
Taveira Blomenhofer, A & Laurent, M 2025, 'Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems', SIAM Journal on Optimization.