有限群群代数中的埃尔米特平方稀疏和
Sparse sum of Hermitian squares in group algebras of finite groups
AI总结：
本文针对有限群群代数中非负元素的稀疏埃尔米特平方和表示，提出阈值分层法并证明其误差指数衰减，为稀疏非交换正性证书提供新算法工具。
摘要：
群代数中的非负元素在调和分析、算子代数与计算优化中处于核心地位。本文研究有限群群代数中非负元素的稀疏埃尔米特平方和（SOHS）表示。我们证明稀疏SOHS问题的凸松弛存在闭式解，即给定元素的平方根。基于该结果，我们提出阈值分层法以近似稀疏SOHS表示。我们针对两种自然残差分析该分层法的误差，建立指数衰减率。值得注意的是，其中一个误差界与群阶无关；当群为循环群或二面体群时，另一个误差界也与群大小无关。这些结果将阿贝尔群的傅里叶平方和现有工作扩展到更广泛的有限群类，并为稀疏非交换正性证书提供新的算法工具。
英文摘要：
Nonnegative elements in group algebras play a central role in harmonic analysis, operator algebras, and computational optimization. This paper investigates sparse sum-of-Hermitian-squares (SOHS) representations of nonnegative elements in the group algebras of finite groups. We prove that the convex relaxation of the sparse SOHS problem admits a closed-form solution, namely the square root of the given element. Based on this result, we propose a thresholding hierarchy for approximating sparse SOHS representations. We analyze the error of this hierarchy with respect to two natural residuals and establish exponential decay rates. Remarkably, one error bound is independent of the group order, and the other is also group-size independent when the group is cyclic or dihedral. These results extend existing work on Fourier sums of squares for abelian groups to a broader class of finite groups and provide new algorithmic tools for sparse noncommutative positivity certificates.
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