Text
Low-Rank Univariate Sum of Squares Has No Spurious Local Minima
We study the problem of decomposing a polynomial p into a sum of r squares by minimizing a quadratically penalized objective f p(u) = || ∑r i=1 u2i-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.
Barcode | Tipe Koleksi | Nomor Panggil | Lokasi | Status | |
---|---|---|---|---|---|
art147445 | null | Artikel | Gdg9-Lt3 | Tersedia namun tidak untuk dipinjamkan - No Loan |
Tidak tersedia versi lain