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.

翻译：我们研究通过最小化二次惩罚目标 $f_p(\mathbf{u}) = \left\lVert \sum_{i=1}^r u_i^2 - p\right\lVert^2$ 将多项式 $p$ 分解为 $r$ 个平方和的问题。该目标是非凸的，等价于编码平方和分解的半定规划（SDP）的秩-$r$ Burer-Monteiro 分解。我们证明，对于所有单变量多项式 $p$，若 $r \ge 2$，则 $f_p(\mathbf{u})$ 不存在虚假的二阶临界点，即所有局部最优解也是全局最优解。这与先前研究形成对比，该研究表明对于一般 SDP，除一般性条件外，$r$ 需大致等于约束数量（即 $p$ 的次数）的平方根，才能避免虚假二阶临界点。我们的证明使用了计算代数几何的工具，可被理解为利用一阶和二阶必要条件构造了一个证书。此外，我们表明通过选择基于圆上等距采样点的范数，梯度 $\nabla f_p$ 可利用快速傅里叶变换以近线性时间计算。实验表明，该方法在使用 L-BFGS 等一阶优化算法时具有极快的收敛速度，且可实现近线性扩展到百万次多项式。