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

TL;DR

This paper analyzes convergence rates of two hierarchies for polynomial optimization on the sphere, linking moment-sos and spectral bounds, and introduces a new de Finetti theorem with explicit constants.

Contribution

It establishes the limits of spectral bounds convergence, introduces a novel banded de Finetti theorem, and improves existing bounds using the polynomial kernel method.

Findings

Spectral bounds cannot converge faster than O(1/r^2).

The paper introduces a new banded de Finetti theorem for real matrices.

Improves the convergence rate for real maximally symmetric matrices to O(1/r^2).

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).