Approximating Nonnegative Polynomials via Spectral Sparsification

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 show that for any fixed $m \geq 3$, all linear $m$-dimensional sections of the nonnegative cone that include $(x_1^2+x_2^2+\cdots + x_n^2)^d$ have a constant 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 the Kadison--Singer problem. We also discuss a randomized polyhedral approximation which might be of independent interest.