Higher Interpolation and Extension for Persistence Modules
Higher Interpolation and Extension for Persistence Modules
复制标题
持久模块的更高插值和扩展
DOI：
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发表时间：
2016
影响因子：
1.2
通讯作者：
Vidit Nanda
中科院分区：
文献类型：
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作者：
Peter Bubenik;V. Silva;Vidit Nanda
拓扑持久性在当代数据分析中的使用为持久性模块空间的几何和功能分析结构的研究提供了相当大的动力。在本文中，我们隔离的连贯性标准，保证了非扩张映射到这个空间的嵌入域到更大的环境度量空间的可扩展性。我们的一致性标准是范畴理论，允许Kan扩展提供所需的扩展。我们的主要建设给出了一个等距嵌入的度量空间到度量空间的持久性模块的时空值在这个度量空间。由于这样的“更高的插值”，它成为可能的比较Vietoris-Rips和v{C}ech复合物内建的空间持久性模块。
The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion which guarantees the extensibility of non-expansive maps into this space across embeddings of the domain to larger ambient metric spaces. Our coherence criterion is category-theoretic, allowing Kan extensions to provide the desired extensions. Our main construction gives an isometric embedding of a metric space into the metric space of persistence modules with values in the spacetime of this metric space. As a consequence of such "higher interpolation", it becomes possible to compare Vietoris-Rips and v{C}ech complexes built within the space of persistence modules.