交织的相关理论
A Relative Theory of Interleavings
摘要：
交错距离，虽然最初是为持久同调而开发的，但已经被推广到测量在许多偏序集甚至小范畴上建模的函子之间的距离。现有的理论要求这样的偏序集具有超线性平移族或类似的结构。然而，许多拓扑数据分析感兴趣的偏序集，如zig-zag偏序集和胞腔复形的面关系偏序集，不允许有趣的平移，因此不承认一个很好的交织理论。在本文中，我们展示了如何可以侧步这个限制提供了一个一般的理论，其中一个映射到偏序集，不承认有趣的翻译，如格的下集，然后定义交错相对于这个地图。我们的理论的一部分包括一个严格的概念离散化或“像素化”的偏序集模块，这反过来又是我们使用的交错推理。我们提供了一个近似条件，在设置的晶格产生两种可能的像素化，这两种都是保证在交织距离接近。最后，我们考虑了度量空间上余层的交织推理，并给出了欧几里得空间上网格结构上的交织的一个显式描述。
英文摘要：
The interleaving distance, although originally developed for persistent homology, has been generalized to measure the distance between functors modeled on many posets or even small categories. Existing theories require that such a poset have a superlinear family of translations or a similar structure. However, many posets of interest to topological data analysis, such as zig-zag posets and the face relation poset of a cell-complex, do not admit interesting translations, and consequently don't admit a nice theory of interleavings. In this paper we show how one can side-step this limitation by providing a general theory where one maps to a poset that does admit interesting translations, such as the lattice of down sets, and then defines interleavings relative to this map. Part of our theory includes a rigorous notion of discretization or "pixelization" of poset modules, which in turn we use for interleaving inference. We provide an approximation condition that in the setting of lattices gives rise to two possible pixelizations, both of which are guaranteed to be close in the interleaving distance. Finally, we conclude by considering interleaving inference for cosheaves over a metric space and give an explicit description of interleavings over a grid structure on Euclidean space.
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