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arXiv · 1612.06245

Approximating Nonnegative Polynomials via Spectral Sparsification

Alperen A. Ergür
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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.

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Alperen A. Ergür. 2016-12-19. Approximating Nonnegative Polynomials via Spectral Sparsification. https://doi.org/10.1137/17m1121743

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