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arXiv:2406.09506 · Linear Algebra and its Applications

The polarization hierarchy for polynomial optimization over convex bodies, with applications to nonnegative matrix rank

Martin Plávala, Laurens T. Ligthart, David Gross

Published May 22, 2025• 1 citation
arXivHTMLPDF

Abstract

We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy , which has previously been introduced for the study of polynomial optimization problems over state spaces of C ⁎ ⁎ -algebras, to convex cones in finite dimensions. If the convex bodies can be characterized by linear or semidefinite programs, then the same is true for our hierarchy. Convergence is proven by relating the problem to a certain de Finetti theorem for general probabilistic theories , which are studied as possible generalizations of quantum mechanics . We apply the method to the problem of nonnegative matrix factorization, and in particular to the nested rectangles problem . A numerical implementation of the third level of the hierarchy is shown to give rise to a very tight approximation for this problem.

DOI
10.1016/j.laa.2025.05.019

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Comment on "The polarization hierarchy for polynomial optimization over convex bodies, with applications to nonnegative matrix rank"

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We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy , which has previously been introduced for the study of polynomial optimization problems over state spaces of C ⁎ ⁎ -algebras, to convex cones in finite dimensions. If the convex bodies can be characterized by linear or semidefinite programs, then the same is true for our hierarchy. Convergence is proven by relating the problem to a certain de Finetti theorem for general probabilistic theories , which are studied as possible generalizations of quantum mechanics . We apply the method to the problem of nonnegative matrix factorization, and in particular to the nested rectangles problem . A numerical implementation of the third level of the hierarchy is shown to give rise to a very tight approximation for this problem.
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Abstract We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form Tr[H(D⊗E)]{\mathrm {Tr}}\big [H(D\otimes E)\big ]Tr[H(D⊗E)] Tr [ H ( D ⊗ E ) ] , maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo’s asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brandão and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC’12, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from O(dk/2)O(d^{k/2})O(dk/2) O ( d k / 2 ) to poly(d,k){\mathrm {poly}}(d,k)poly(d,k) poly ( d , k ) , where d denotes local dimension and k the number of copies.
We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form Tr[M(X⊗Y)]\mathrm{Tr}\big[M(X\otimes Y)\big]Tr[M(X⊗Y)], maximized with respect to semidefinite constraints on XXX and YYY. Applied to the problem of quantum error correction this gives hierarchies of efficiently computable outer bounds on the optimal fidelity for any message dimension and error model. The first level of our hierarchies corresponds to the non-signalling assisted fidelity previously studied by [Leung & Matthews, IEEE Trans.~Inf.~Theory 2015], and positive partial transpose constraints can be added and used to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo's asymptotic de Finetti theorem for quantum channels. Finally, our proof methods also allow us to answer an open question from [Brand\~ao & Harrow, STOC 2013] by improving the approximation factor of de Finetti theorems with no symmetry from O(dk/2)O(d^{k/2})O(dk/2) to poly(d,k)\mathrm{poly}(d,k)poly(d,k), where ddd denotes local dimension and kkk the number of copies.
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 [Formula: see text] for the [Formula: see text]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 [Formula: see text], 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 [Formula: see text] 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 [Formula: see text] for real maximally symmetric matrices, improving an earlier result in [Formula: see text] of Doherty and Wehner (2012).