Convergence of SDP hierarchies for polynomial optimization on the hypersphere

TL;DR

This paper establishes bounds on the accuracy of SDP relaxations for polynomial optimization on the hypersphere, leveraging quantum de Finetti theorems to connect moment matrices with approximate measures.

Contribution

It introduces a novel de Finetti theorem for symmetric matrices to analyze the convergence of SDP hierarchies in polynomial optimization.

Findings

Bounded the accuracy of SDP relaxations.

Connected moment matrices to approximate measures.

Provided theoretical guarantees for optimization on the hypersphere.

Abstract

We show how to bound the accuracy of a family of semi-definite programming relaxations for the problem of polynomial optimization on the hypersphere. Our method is inspired by a set of results from quantum information known as quantum de Finetti theorems. In particular, we prove a de Finetti theorem for a special class of real symmetric matrices to establish the existence of approximate representing measures for moment matrix relaxations.