论文标题
非线性低率矩阵恢复问题的全球和本地分析
Global and Local Analyses of Nonlinear Low-Rank Matrix Recovery Problems
论文作者
论文摘要
限制的等轴测特性（RIP）是一种众所周知的条件，可确保在低级别基质恢复问题中没有线性测量的局部局部最小值。在本文中，我们介绍了一种名为“绑定差异”特性（BDP）的新型属性，以研究非线性测量的低率矩阵恢复问题。我们首先使用RIP和BDP共同关注Rank-1矩阵恢复问题，为此，我们提出了一个新标准，以证明整个空间中虚假的本地最小值不存在。然后，我们以任意等级分析一般案例，并得出一种条件，以排除在真实解决方案围绕球中具有伪造解决方案的可能性。与RIP上的现有界限相比，开发的条件可实现更强的理论保证。
The restricted isometry property (RIP) is a well-known condition that guarantees the absence of spurious local minima in low-rank matrix recovery problems with linear measurements. In this paper, we introduce a novel property named bound difference property (BDP) to study low-rank matrix recovery problems with nonlinear measurements. Using RIP and BDP jointly, we first focus on the rank-1 matrix recovery problem, for which we propose a new criterion to certify the nonexistence of spurious local minima over the entire space. We then analyze the general case with an arbitrary rank and derive a condition to rule out the possibility of having a spurious solution in a ball around the true solution. The developed conditions lead to much stronger theoretical guarantees than the existing bounds on RIP.
