In this paper, we study a general low-rank matrix recovery problem with linear measurements corrupted by some noise. The objective is to understand under what conditions on the restricted isometry property (RIP) of the problem local search methods can find the ground truth with a small error. By analyzing the landscape of the non-convex problem, we first propose a global guarantee on the maximum distance between an arbitrary local minimizer and the ground truth under the assumption that the RIP constant is smaller than $1/2$. We show that this distance shrinks to zero as the intensity of the noise reduces. Our new guarantee is sharp in terms of the RIP constant and is much stronger than the existing results. We then present a local guarantee for problems with an arbitrary RIP constant, which states that any local minimizer is either considerably close to the ground truth or far away from it. Next, we prove the strict saddle property, which guarantees the global convergence of the perturbed gradient descent method in polynomial time. The developed results demonstrate how the noise intensity and the RIP constant of the problem affect the landscape of the problem.

翻译：本文研究通用线性测量被噪声污染的低秩矩阵恢复问题。旨在理解在何种受限等距性质条件下，局部搜索方法能够以较小误差逼近真实解。通过分析非凸问题的景观，我们首先提出全局性保证：当受限等距性质常数小于$1/2$时，任意局部极小值与真实解的最大距离可被界定。研究表明该距离随噪声强度降低而趋于零。这一新保证在受限等距性质常数意义上具有尖锐性，且显著优于现有结果。随后针对任意受限等距性质常数的问题提出局部性保证：任何局部极小值要么非常接近真实解，要么远离真实解。进一步证明严格鞍点性质，确保扰动梯度下降法能在多项式时间内全局收敛。这些结果揭示了噪声强度与受限等距性质常数对问题景观的影响机制。