通过优化原理实现群的单轨道恢复
Single-Orbit Recovery of Groups via an Optimization Principle
浏览论文内容
中文总结 AI 辅助
本文提出通过优化原理从向量轨道恢复有限群结构，给出正交矩阵族成为右正则表示的充分条件，并在D4和A4上验证了方法的有效性。
中文摘要 AI 辅助
本文的目标是基于一个向量在群作用下的轨道来识别有限群。当有限群在希尔伯特空间上酉作用时，相应的格拉姆矩阵位于群代数中，并满足编码代数群结构的恒等式。我们探讨其逆问题：给定由群作用下向量轨道得到的格拉姆矩阵，能否恢复底层的抽象群结构？我们的主要结果确定了正交矩阵族的一组简短条件，这些条件强制该族成为有限群的右正则表示。基于这一刻画，我们开发了一个分阶段优化框架，逐步施加这些条件。在二面体群$D_4$和四面体旋转群$A_4$上的数值实验精确恢复了$D_4$，并以适度精度恢复了全部十二个$A_4$矩阵。
英文摘要
The objective of this paper is to identify a finite group based on the orbit of a vector under its action. When a finite group acts unitarily on a Hilbert space, the associated Gram matrix lies in the group algebra and satisfies identities that encode the algebraic group structure. We ask about the converse: Given a Gram matrix obtained from the orbit of a vector under the group action, can one recover the underlying \emph{abstract} group structure? Our main result identifies a short list of conditions on a family of orthogonal matrices that force the family to be the right regular representation of a finite group. Building on this characterization, we develop a staged optimization framework that enforces these conditions progressively. Numerical experiments on the dihedral group $D_4$ and the tetrahedral rotation group $A_4$ recover $D_4$ exactly and all twelve $A_4$ matrices up to modest precision.
发表机构
- University of Houston（休斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。
