Sum-of-Squares Optimization and the Sparsity Structure of Equiangular Tight Frames
Sum-of-Squares Optimization and the Sparsity Structure of Equiangular Tight Frames
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DOI：
10.1109/sampta45681.2019.9030987
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发表时间：
2019-01
期刊：
影响因子：
--
通讯作者：
A. Bandeira;Dmitriy Kunisky
中科院分区：
文献类型：
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作者：
A. Bandeira;Dmitriy Kunisky
等角紧框架（ETF）可用于构造平方和（SOS）优化中产生的半定规划的可行点的示例。我们展示了作者最近探讨这种联系的工作中的计算如何推广，也为（真实的和复杂的）ETF 的稀疏性产生了新的界限。一个推论表明，在控制合成矩阵的不同行的稀疏模式之间的重叠的矩阵不等式中实现紧密性的意义上，对应于有限射影平面的 Steiner ETF 是最优稀疏的。我们还提出了一些有关我们技术进一步推广的自然开放问题。
Equiangular tight frames (ETFs) may be used to construct examples of feasible points for semidefinite programs arising in sum-of-squares (SOS) optimization. We show how generalizing the calculations in a recent work of the authors’ that explored this connection also yields new bounds on the sparsity of (both real and complex) ETFs. One corollary shows that Steiner ETFs corresponding to finite projective planes are optimally sparse in the sense of achieving tightness in a matrix inequality controlling overlaps between sparsity patterns of distinct rows of the synthesis matrix. We also formulate several natural open problems concerning further generalizations of our technique.