论文标题
低相位近似
Low Phase-Rank Approximation
论文作者
论文摘要
在本文中，我们提出并解决了一个低相位近似问题，这与众所周知的低级近似问题和Schmidt-Mirsky定理相当。更具体地说，可以通过其增益和阶段来指定非零的复合数，尽管通常可以接受矩阵的收益可以由其单数值定义，但对其阶段没有广泛接受的定义。在这项工作中，我们考虑了部门矩阵，其数值范围不包含起源，并采用诸如阶段等矩阵的规范角度。与定义为其非零奇异值的数量的矩阵的等级类似，我们将部门矩阵的相位级定义为其非零阶段的数量。虽然低级别近似问题与矩阵算术均值有关，但作为自然平行的矩阵算法，我们使用矩阵几何手段来制定低相位近似问题，以测量近似误差。然后，当客观矩阵和大约限制为正疑问时，就获得了提出问题的解决方案的表征。此外，获得的溶液具有与施密特 - 米尔斯基定理相同的近似近似问题的风味。此外，我们使用部门矩阵之间的大地距离提供了低相位近似问题的替代公式。当涉及矩阵还被假定为单一时，这两个配方会产生完全相同的解决方案。
In this paper, we propose and solve a low phase-rank approximation problem, which serves as a counterpart to the well-known low-rank approximation problem and the Schmidt-Mirsky theorem. More specifically, a nonzero complex number can be specified by its gain and phase, and while it is generally accepted that the gains of a matrix may be defined by its singular values, there is no widely accepted definition for its phases. In this work, we consider sectorial matrices, whose numerical ranges do not contain the origin, and adopt the canonical angles of such matrices as their phases. Similarly to the rank of a matrix defined to be the number of its nonzero singular values, we define the phase-rank of a sectorial matrix as the number of its nonzero phases. While a low-rank approximation problem is associated with matrix arithmetic means, as a natural parallel we formulate a low phase-rank approximation problem using matrix geometric means to measure the approximation error. A characterization of the solutions to the proposed problem is then obtained, when both the objective matrix and the approximant are restricted to be positive-imaginary. Moreover, the obtained solution has the same flavor as the Schmidt-Mirsky theorem on low-rank approximation problems. In addition, we provide an alternative formulation of the low phase-rank approximation problem using geodesic distances between sectorial matrices. The two formulations give rise to the exact same set of solutions when the involved matrices are additionally assumed to be unitary.
