Role of sparsity and structure in the optimization landscape of non-convex matrix sensing
Role of sparsity and structure in the optimization landscape of non-convex matrix sensing
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DOI：
10.1007/s10107-020-01590-2
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发表时间：
2020-11
影响因子：
2.7
通讯作者：
Igor Molybog;S. Sojoudi;J. Lavaei
中科院分区：
文献类型：
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作者：
Igor Molybog;S. Sojoudi;J. Lavaei
在这项工作中，我们研究了优化景观的非凸矩阵传感问题，已知有许多局部极小值在最坏的情况下。由于现有的结果与限制等距属性（RIP）的概念，不能直接捕捉一个给定的问题的底层结构，他们很难被应用到现实世界中的问题，数据量不是过高。为了解决这个问题，我们开发的概念的核结构属性，以获得必要和充分条件的虚假的局部解的任何类别的矩阵传感问题在给定的搜索空间。这个概念精确地捕捉了问题的基本稀疏性和结构，基于圆锥优化中的工具。我们简化了某类问题的条件，以显示其满意度，并将其应用于电力系统的数据分析。
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.