Frame Potentials and the Geometry of Frames
Frame Potentials and the Geometry of Frames
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DOI：
10.1007/s00041-015-9408-z
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发表时间：
2014-07
影响因子：
1.2
通讯作者：
B. Bodmann;John I. Haas
中科院分区：
文献类型：
--
作者：
B. Bodmann;John I. Haas
本文讨论了框架势优化器的几何结构。我们考虑有限、实或复框架和旋转或酉不变势，主要研究Parseval框架，这意味着要优化的框架势是属于有限Parseval框架的Gram矩阵流形上的函数。在已知的等范数类和等角Parseval框架的基础上，我们引入了等分布Parseval框架，它们比等角型框架更一般，但比等范数框架具有更多的结构。我们还给出了这类与Grassman框架重合的例子，Grassman框架是框架向量之间内积的最大幅值的最小化。这些不同类型的框架的特征与框架势的优化有关。基于Łojasiewicz的结果，我们证明了属于Parseval框架的Gram矩阵流形上的实解析框架势的梯度下降总是收敛到一个临界点。然后，我们推导出与不同框架势选择的临界点相关的几何结构。从而证明了这类势能族的最优框架是等范数的，或附加等分的，甚至是等分布的。
This paper concerns the geometric structure of optimizers for frame potentials. We consider finite, real or complex frames and rotation or unitarily invariant potentials, and mostly specialize to Parseval frames, meaning the frame potential to be optimized is a function on the manifold of Gram matrices belonging to finite Parseval frames. Next to the known classes of equal-norm and equiangular Parseval frames, we introduce equidistributed Parseval frames, which are more general than the equiangular type but have more structure than equal-norm ones. We also provide examples where this class coincides with that of Grassmannian frames, the minimizers for the maximal magnitude among inner products between frame vectors. These different types of frames are characterized in relation to the optimization of frame potentials. Based on results by Łojasiewicz, we show that the gradient descent for a real analytic frame potential on the manifold of Gram matrices belonging to Parseval frames always converges to a critical point. We then derive geometric structures associated with the critical points of different choices of frame potentials. The optimal frames for families of such potentials are thus shown to be equal-norm, or additionally equipartitioned, or even equidistributed.