Persistent Symmetries in Data Science
演讲者
Jian Liu
时间
2025年11月06日 14:00 至 15:00
地点
A3-4-101
线上
Zoom 928 682 9093
(BIMSA)
摘要
Symmetry is a fundamental concept in understanding both natural phenomena and mathematical structures. This topic develops a comprehensive theory for analyzing the persistent symmetries and degrees of asymmetry in finite point configurations within metric spaces. Using category theory and span categories, we define persistent symmetry groups and introduce new invariants, namely symmetry barcodes and polybarcodes, which capture the birth, death, persistence, and reappearance of symmetries as parameters evolve. We establish metrics and stability theorems for these invariants. The concept of symmetry types is formalized through the action of isometry groups on configuration spaces. To quantitatively characterize symmetry and asymmetry, we introduce measures such as the degree of symmetry and symmetry defect, the latter establishing connections with approximate group theory in Euclidean spaces.