This paper is concerned with low-rank matrix optimization, which has found a wide range of applications in machine learning. This problem in the special case of matrix sensing has been studied extensively through the notion of Restricted Isometry Property (RIP), leading to a wealth of results on the geometric landscape of the problem and the convergence rate of common algorithms. However, the existing results can handle the problem in the case with a general objective function subject to noisy data only when the RIP constant is close to 0. In this paper, we develop a new mathematical framework to solve the above-mentioned problem with a far less restrictive RIP constant. We prove that as long as the RIP constant of the noiseless objective is less than $1/3$, any spurious local solution of the noisy optimization problem must be close to the ground truth solution. By working through the strict saddle property, we also show that an approximate solution can be found in polynomial time. We characterize the geometry of the spurious local minima of the problem in a local region around the ground truth in the case when the RIP constant is greater than $1/3$. Compared to the existing results in the literature, this paper offers the strongest RIP bound and provides a complete theoretical analysis on the global and local optimization landscapes of general low-rank optimization problems under random corruptions from any finite-variance family.

翻译：本文研究低秩矩阵优化问题，该问题在机器学习领域具有广泛应用。矩阵感知这一特殊情形已通过受限等距性质（RIP）得到广泛研究，获得了关于问题几何景观和常见算法收敛速率的丰富成果。然而，当优化目标函数为一般形式且数据含有噪声时，现有结果仅在RIP常数接近0的情况下有效。本文提出一种新的数学框架，可在远不严格的RIP常数条件下解决上述问题。我们证明：当无噪声目标函数的RIP常数小于1/3时，含噪优化问题的任何虚假局部解必接近真实解。通过严格鞍点性质的分析，我们还证明可在多项式时间内找到近似解。针对RIP常数大于1/3的情形，我们刻画了真实解局部邻域内虚假局部极小值的几何结构。与现有文献结果相比，本文给出了最强的RIP界，并针对任意有限方差族随机污染下的广义低秩优化问题，提供了完整的全局与局部优化景观理论分析。