用有限拓扑空间构建新的持久同调框架，避免复杂包含关系。
Persistent Homology via Finite Topological Spaces
- 基于有限度量空间构造拓扑滤链，通过序集与序复形实现函子化转换
- 在密度引导构造下，对输入度量扰动保持持久图的稳定性
- 无需复形间的包含关系，适合处理真实数据中的局部结构
我们提出一种基于有限拓扑空间及其相关序集的持久同调函子框架。从有限度量空间出发，构造一系列有限拓扑的滤链，其结构映射为连续恒等映射。通过函子性地过渡到序集和序复形，得到无需复形间包含关系的持久模。我们证明了序集层面的标准简化操作可保持持久不变量，并在基本密度驱动实例中建立了对输入度量扰动的稳定性，说明稳定性论证自然出现在该框架中。进一步提出一种具体密度引导构造，旨在忠实保留各尺度下的邻域锚点结构，并通过真实数据集上的实现展示了其实际可行性。
原文摘要 · Abstract (English)
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are continuous identity maps. By passing functorially to posets and to order complexes, we obtain persistence modules without requiring inclusion relations between the resulting complexes. We show that standard poset-level simplifications preserve persistent invariants and establish stability of the resulting persistence diagrams under perturbations of the input metric in a basic density-based instantiation, illustrating how stability arguments arise naturally in our framework. We further introduce a concrete density-guided construction, designed to be faithful to anchor neighborhood structure at each scale, and demonstrate its practical viability through an implementation tested on real datasets.
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