Approximating Nonnegative Polynomials via Spectral Sparsification

TL;DR

This paper investigates the complexity of polyhedral approximations to the cone of nonnegative polynomials, revealing exponential facet requirements for general approximations and polynomial-sized sections, using convex geometry and Kadison-Singer results.

Contribution

It establishes exponential lower bounds for general polyhedral approximations and provides efficient approximations for certain sections of the nonnegative polynomial cone.

Findings

Any constant ratio polyhedral approximation requires exponentially many facets in n.

All linear m-dimensional sections including the sum of squares polynomial have polynomial-sized approximations.

The approach leverages convex geometry and recent solutions to the Kadison-Singer problem.

Abstract

We study polyhedral approximations to the cone of nonnegative polynomials. We show that any constant ratio polyhedral approximation to the cone of nonnegative degree $2d$ forms in $n$ variables has to have exponentially many facets in terms of $n$. We also showthat for fixed $m \geq 3$, all linear $m$ dimensional sections of the nonnegative cone that include $(x_1^2+x_2^2+\ldots + x_n^2)^d$ has a costant ratio polyhedral approximation with $O(n^{m-2})$ many facets. Our approach is convex geometric, and parts of the argument rely on the recent solution of Kadison-Singer problem. We also discuss a randomized polyhedral approximation which might be of independent interest.