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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 fp(u)=‖‖∑ri=1u2i−p‖‖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≥2 then fp(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 ∇fp 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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Tight Bounds for Volumetric Spanners and Applications
Given a set of points of interest, a volumetric spanner is a subset of the points using which all the points can be expressed using “small” coefficients (measured in an appropriate norm). This notion, which has also been referred to as a well-conditioned basis, has found several applications, including bandit linear optimization, determinant maximization, and matrix low rank approximation. In this paper, we give almost optimal bounds on the size of volumetric spanners for all L_p norms, and show that they can be constructed using a simple local search procedure. We then show the applications of our result to other tasks and in particular the problem of finding coresets for the Minimum Volume Enclosing Ellipsoid (MVEE) problem.
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- Award ID(s):
- 2008688
- PAR ID:
- 10555344
- Editor(s):
- Oh, A; Naumann, T; Globerson, A; Saenko, K; Hardt, M; Levine, S
- Publisher / Repository:
- Neural Information Processing Systems
- Date Published:
- Subject(s) / Keyword(s):
- Volumetric spanners, Coresets
- Format(s):
- Medium: X
- Location:
- New Orleans, USA
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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