In this work, we study the optimization landscape of the non-convex matrix sensing problem that is known to have many local minima in the worst case. Since the existing results are related to the notion of restricted isometry property (RIP) that cannot directly capture the underlying structure of a given problem, they can hardly be applied to real-world problems where the amount of data is not exorbitantly high. To address this issue, we develop the notion of kernel structure property to obtain necessary and sufficient conditions for the inexistence of spurious local solutions for any class of matrix sensing problems over a given search space. This notion precisely captures the underlying sparsity and structure of the problem, based on tools in conic optimization. We simplify the conditions for a certain class of problems to show their satisfaction and apply them to data analytics for power systems.

中文翻译：

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稀疏性和结构在非凸矩阵传感优化景观中的作用

在这项工作中，我们研究了已知在最坏情况下具有许多局部最小值的非凸矩阵感知问题的优化前景。由于现有结果与不能直接捕获给定问题的底层结构的受限等距特性 (RIP) 的概念有关，因此它们很难应用于数据量不是过高的现实世界问题。为了解决这个问题，我们开发了核结构属性的概念，以获得在给定搜索空间上任何类别的矩阵感知问题不存在虚假局部解的必要和充分条件。这个概念基于圆锥优化中的工具，精确地捕捉了问题的潜在稀疏性和结构。