论文标题
有限群体的组代数中的Hermitian广场的稀疏总和
Sparse sum of Hermitian squares in group algebras of finite groups
论文作者
论文摘要
组代数中的非负元素在功能，措施和操作员的研究中起着至关重要的作用。本文重点介绍了有限群体代数组中非负元素的Hermitian广场（SOHS）的总和。我们首先证明，对于给定的非负元素，稀疏SOHS问题的凸松弛的最佳解完全是其平方根。然后，我们为稀疏的SOHS问题提出了一个层次结构，并分析了层次结构相对于两种类型的残差的误差。值得注意的是，我们证明这两个错误呈指数衰减。此外，我们表明，对于一种错误，衰减率与组的顺序无关。对于另一种类型，我们证明了速率也独立于组顺序，只要组是循环或二面的。
Non-negative elements in group algebras play a crucial role in the study of functions, measures and operators. This paper focuses on the sum of Hermitian squares (SOHS) of non-negative elements in group algebras of finite groups. We first prove that for a given non-negative element, the optimal solution of the convex relaxation of the sparse SOHS problem is precisely its square root. Then we propose a hierarchy for the sparse SOHS problem, and we analyze the error of the hierarchy with respect to two types of residuals. Notably, we prove that both errors decay exponentially. Moreover, we show that for one type of error, the decay rate is independent of the order of the group. For the other type, we demonstrate that the rate is also independent of the group order, provided that the group is cyclic or dihedral.
