论文标题
广义持久图的空间的拓扑和度量特性
Topological and metric properties of spaces of generalized persistence diagrams
论文作者
论文摘要
由持续的同源性和拓扑数据分析的动机，我们考虑了具有杰出子集的度量空间上的正式总和。这些正式的总和我们称为持久图，具有一个规范的1参数指标家族，称为Wasserstein距离。我们研究这些空间的拓扑和度量特性。即使在半平面上持续图的情况下，我们的一些结果也是新的。在轻度条件下，没有持久图具有紧凑的邻居。如果基本的度量空间为$σ$ - compact，那么持久图的空间也是如此。但是，在轻度条件下，持久图的空间不是半体体的，并且从该空间到拓扑空间的功能空间是不可分割的。持久图的空间继承了与基础度量空间的完整性和可分离性。某些持久图的空间继承了路径连接，是一个长度空间，并且是一个大地的空间，但其他空间则没有。我们给出了一组持久图的标准，使其完全有限且相对紧凑。我们还研究了持久图的空间的曲率和尺寸及其嵌入到希尔伯特空间中。作为具有独立感兴趣的重要技术步骤，我们为存在持续图的最佳匹配提供了必要的条件。
Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics called Wasserstein distances. We study the topological and metric properties of these spaces. Some of our results are new even in the case of persistence diagrams on the half-plane. Under mild conditions, no persistence diagram has a compact neighborhood. If the underlying metric space is $σ$-compact then so is the space of persistence diagrams. However, under mild conditions, the space of persistence diagrams is not hemicompact and the space of functions from this space to a topological space is not metrizable. Spaces of persistence diagrams inherit completeness and separability from the underlying metric space. Some spaces of persistence diagrams inherit being path connected, being a length space, and being a geodesic space, but others do not. We give criteria for a set of persistence diagrams to be totally bounded and relatively compact. We also study the curvature and dimension of spaces of persistence diagrams and their embeddability into a Hilbert space. As an important technical step, which is of independent interest, we give necessary and sufficient conditions for the existence of optimal matchings of persistence diagrams.
