A topological study of functional data and Frechet functions of metric measure spaces
A topological study of functional data and Frechet functions of metric measure spaces
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度量测度空间函数数据和Frechet函数的拓扑研究
DOI：
10.1007/s41468-019-00037-8
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Mio, W.
中科院分区：
文献类型：
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作者：
Hang, H;Memoli, F;Mio, W.
我们研究紧致拓扑空间上的函数数据和表示为紧致度量测度空间的结构数据的持久同源性。我们的目标之一是定义持久同源性，以便捕获信号形状的主要属性，消除其他高度持久的同源性类别，这些同源性类别可能仅仅因为定义信号的域的性质而存在。我们使用淡化信号弱区域的指标来研究这些不变量的稳定性。如果两个信号在较强的区域表现出高度相似性，则它们之间的距离很小，无论其完整域的性质如何，特别是允许不同的同伦类型。在此框架内研究了数据度量测量空间的持久同源性的一致性和估计。我们还将该方法应用于通过从热核导出的度量松弛来构建紧凑黎曼流形数据的多尺度拓扑描述符。
We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties of the shape of a signal, eliminating otherwise highly persistent homology classes that may exist simply because of the nature of the domain on which the signal is defined. We investigate the stability of these invariants using metrics that downplay regions where signals are weak. The distance between two signals is small if they exhibit high similarity in regions where they are strong, regardless of the nature of their full domains, in particular allowing different homotopy types. Consistency and estimation of persistent homology of metric measure spaces from data are studied within this framework. We also apply the methodology to the construction of multi-scale topological descriptors for data on compact Riemannian manifolds via metric relaxations derived from the heat kernel.