ORTHOGONAL TRACE-SUM MAXIMIZATION: TIGHTNESS OF THE SEMIDEFINITE RELAXATION AND GUARANTEE OF LOCALLY OPTIMAL SOLUTIONS.
ORTHOGONAL TRACE-SUM MAXIMIZATION: TIGHTNESS OF THE SEMIDEFINITE RELAXATION AND GUARANTEE OF LOCALLY OPTIMAL SOLUTIONS.
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正交迹和最大化：半定松弛的严格性和局部最优解的保证。
DOI：
10.1137/21m1422707
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Zhou,Hua
中科院分区：
文献类型：
--
作者：
Won,Joong-Ho;Zhang,Teng;Zhou,Hua
本文研究了半正交矩阵中矩阵二次型的迹和的优化问题，它可以看作是旋转同步的推广。当问题是非凸的时，本文证明了它的半定规划松弛在具有小噪声的数量级的加性噪声模型下以高概率精确地解决了原始的非凸问题。此外，本文还证明了文[SIAM J.Matrix anal]中所考虑的全局最优性的充分条件是大概率的。[参考文献2(2021)，第859-882页]在类似的小噪声条件下也是必要的。这些结果可以看作是已有的相位同步结果的推广。
This paper studies an optimization problem on the sum of traces of matrix quadratic forms insemiorthogonal matrices, which can be considered as a generalization of the synchronization of rotations. While the problem is nonconvex, this paper shows that its semidefinite programming relaxation solves the original nonconvex problems exactly with high probability under an additive noise model with small noise in the order of. In addition, it shows that, with high probability, the sufficient condition for global optimality considered in Won, Zhou, and Lange [SIAM J. Matrix Anal. Appl., 2 (2021), pp. 859--882] is also necessary under a similar small noise condition. These results can be considered as a generalization of existing results on phase synchronization.