空间的尺度几何学 持久性图的空间
Mauricio Che1, Fernando Galaz-García1, Luis Guijarro2
1Department of Mathematical Sciences, Durham University, Durham, UK.
概括
本研究探讨了持久性图的几何性质,在拓数据分析中至关重要. 这项研究引入了函数来分析这些空间,揭示了完整性和曲率等特性,以增强数据解释.
科学领域:
- 拓学的拓学
- 数据分析 数据分析
- 尺度几何几何学 尺度几何学
背景情况:
- 持久性图在拓数据分析 (TDA) 中是基本的.
- 了解持久性图的空间的几何性质对于它们的应用至关重要.
- 之前的工作集中在特定的方面,但需要一个全面的几何框架.
研究的目的:
- 为了研究空间的局部和全球几何性质的持久性图.
- 构建一个函数家族,将二元对映射到指点二元空间.
- 分析这些函数所保留的度量属性和欧几里德持久性图空间的维度.
主要方法:
- 一个函数家族的构造,将尖尖的度量空间分配给度量对 (X,A).
- 关于格罗莫夫-豪斯多夫收的顺序连续性的分析.
- 测量特性的保存 (完整性,可分离性,地质性,亚历山德罗夫曲率) 的研究.
- 在空图中描述方向的度量空间.
- 证明Borel概率测量的Fréchet平均数组的非空性.
主要成果:
- 在格罗莫夫-豪斯多夫收下,函数是顺序连续的.
- 保留了空间的完整性和分离性.
- 维护地质性和非负的亚历山德罗夫曲率对于空间.
- 对于具有有限秒矩和紧支的尺度,Fréchet平均集是非空的.
- 欧几里德持久性图的空间显示无限覆盖,豪斯多夫,非对称,Assouad和Assouad-Nagata维度.
结论:
- 开发的几何框架为持久性图空间的结构提供了新的见解.
- 函数的属性使得我们能够更深入地了解尺度和几何特征.
- 无限维度结果对TDA中的持久性图的计算和理论分析具有重要意义.
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