Certifying the Absence of Spurious Local Minima at Infinity
针对非凸无约束优化问题，本文提出条件来证明无穷远处不存在虚假局部最小值，并验证其在主成分分析、矩阵感知和线性神经网络等应用中的有效性。
When searching for global optima of nonconvex unconstrained optimization problems, it is desirable that every local minimum be a global minimum. This property of having no spurious local minima is true in various problems of interest nowadays, including principal component analysis, matrix sensing, and linear neural networks. However, since these problems are noncoercive, they may yet have spurious local minima at infinity. The classical tools used to analyze the optimization landscape, namely the gradient and the Hessian, are incapable of detecting spurious local minima at infinity. In this paper, we identify conditions that certify the absence of spurious local minima at infinity, one of which is having bounded subgradient trajectories. We check that they hold in several applications of interest.