Weak Decoupling, Polynomial Folds, and Approximate Optimization over the Sphere

TL;DR

This paper develops approximation algorithms for maximizing the absolute value of degree-$d$ homogeneous polynomials over the sphere, using new decoupling tools and sum-of-squares relaxations, with trade-offs between accuracy and computational complexity.

Contribution

It introduces novel decoupling techniques and approximation algorithms that leverage higher level sum-of-squares relaxations for polynomial optimization over the sphere.

Findings

Approximation within factor $O_d((n/q)^{d/2-1})$ in $n^{O(q)}$ time for general polynomials.

Improved approximation for non-negative coefficient polynomials: $O_d((n/q)^{d/4-1/2})$.

Approximation for sparse polynomials with $m$ monomials: $O_d( oot{2}{m/q})$.

Abstract

We consider the following basic problem: given an $n$-variate degree-$d$ homogeneous polynomial $f$ with real coefficients, compute a unit vector $x \in \mathbb{R}^n$ that maximizes $|f(x)|$. Besides its fundamental nature, this problem arises in diverse contexts ranging from tensor and operator norms to graph expansion to quantum information theory. The homogeneous degree $2$ case is efficiently solvable as it corresponds to computing the spectral norm of an associated matrix, but the higher degree case is NP-hard. We give approximation algorithms for this problem that offer a trade-off between the approximation ratio and running time: in $n^{O(q)}$ time, we get an approximation within factor $O_d((n/q)^{d/2-1})$ for arbitrary polynomials, $O_d((n/q)^{d/4-1/2})$ for polynomials with non-negative coefficients, and $O_d(\sqrt{m/q})$ for sparse polynomials with $m$ monomials. The approximation guarantees are with respect to the optimum of the level-$q$ sum-of-squares (SoS) SDP relaxation of the problem. Known polynomial time algorithms for this problem rely on "decoupling lemmas." Such tools are not capable of offering a trade-off like our results as they blow up the number of variables by a factor equal to the degree. We develop new decoupling tools that are more efficient in the number of variables at the expense of less structure in the output polynomials. This enables us to harness the benefits of higher level SoS relaxations. We complement our algorithmic results with some polynomially large integrality gaps, albeit for a slightly weaker (but still very natural) relaxation. Toward this, we give a method to lift a level-$4$ solution matrix $M$ to a higher level solution, under a mild technical condition on $M$.