Low-Rank Univariate Sum of Squares Has No Spurious Local Minima
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We study the problem of decomposing a polynomial p into a sum of r squares by minimizing a quadratically-penalized objective f_p(𝐮) = ‖∑_i=1^r u_i^2 - p‖^2. This objective is non-convex 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 ≥ 2 then f_p(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 ∇ 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.
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