Low-Rank Univariate Sum of Squares Has No Spurious Local Minima

TL;DR

This paper proves that for univariate polynomial sum of squares decompositions with rank at least 2, all local minima are global, ensuring reliable optimization without spurious solutions, and introduces a fast Fourier transform-based computation method.

Contribution

It establishes the absence of spurious local minima in univariate sum of squares problems for rank ≥ 2 and develops an efficient FFT-based gradient computation technique.

Findings

No spurious second-order critical points for rank ≥ 2

Fast Fourier transform enables nearly linear time gradient computation

First-order methods achieve rapid convergence on large-degree polynomials

Abstract

We study the problem of decomposing a polynomial $p$ into a sum of $r$ squares by minimizing a quadratically penalized objective $f_p(\mathbf{u}) = \left\lVert \sum_{i=1}^r u_i^2 - p\right\lVert^2$. This objective is nonconvex and is equivalent to the rank-$r$ Burer-Monteiro factorization of a semidefinite program (SDP) encoding the sum of squares decomposition. We show that for all univariate polynomials $p$, if $r \ge 2$ then $f_p(\mathbf{u})$ has no spurious second-order critical points, showing that all local optima are also global optima. This is in contrast to previous work showing that for general SDPs, in addition to genericity conditions, $r$ has to be roughly the square root of the number of constraints (the degree of $p$) for there to be no spurious second-order critical points. Our proof uses tools from computational algebraic geometry and can be interpreted as constructing a certificate using the first- and second-order necessary conditions. We also show that by choosing a norm based on sampling equally-spaced points on the circle, the gradient $\nabla f_p$ can be computed in nearly linear time using fast Fourier transforms. Experimentally we demonstrate that this method has very fast convergence using first-order optimization algorithms such as L-BFGS, with near-linear scaling to million-degree polynomials.