We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20 (2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31 (3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.

中文翻译：

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从几何和拓扑角度进行层理论分层学习

我们从几何和拓扑的角度研究分层学习的层理论解释。我们的主要结果是构建了分层学习算法，该算法以具有 Alexandroff 拓扑的偏序集上的层为框架。我们证明，由此产生的分解是唯一的最小分层，对于该分层，地层是均质的，并且给定的层是可构造的。特别是，当我们选择使用局部同源层时，我们的算法提供了 Bendich 等人中给出的局部同源转移算法的替代方案。（第 23 届 ACM-SIAM 离散算法研讨会论文集，第 1355-1370 页，ACM，纽约，2012 年）和 Nanda 中给出的上同调分层算法（Found. Comput. Math. 20 (2), 195– 222, 2020)。此外，我们给出了基于 Breiding 等人的几何技术的分层示例。（Rev. Mat. Complut. 31 (3), 545–593, 2018），说明了如何使用层论方法从拓扑和几何角度研究分层。这种方法还通过说明层理论语言在推广现有算法中的效用，指出了层理论在拓扑数据分析研究中的未来应用。