标题： 非各向同性持久同调：利用PH的度量依赖性 显示英文标题

标题： Non-isotropic Persistent Homology: Leveraging the Metric Dependency of PH

摘要： 持久同调是一个广泛使用的拓扑数据分析工具，它基于指定的过滤函数，创建点云拓扑性质的简洁描述。 大多数用于持久同调的过滤函数（隐式地）依赖于一个选定的度量，通常以$\mathbb{R}^n$上的标准欧几里得度量进行选择。 最近的研究尝试使用距离到度量函数来揭示点云上的“真实”度量，以获得更有意义的持久同调结果。 在这里，我们对该问题提出了一种不同的看法：我们认为当将持久同调限制在单一（正确）距离函数时，点云的信息会丢失。 相反，我们展示了如何通过改变底层空间上的距离函数并分析相应的持续图变化，可以提取额外的拓扑和几何信息。 最后，我们通过数值实验表明，非各向同性的持久同调可以准确地提取随机生成点云的方向、取向方差和缩放信息，并在现实世界的数据上进行了一些实验。 显示英文摘要

摘要： Persistent Homology is a widely used topological data analysis tool that creates a concise description of the topological properties of a point cloud based on a specified filtration. Most filtrations used for persistent homology depend (implicitly) on a chosen metric, which is typically agnostically chosen as the standard Euclidean metric on $\mathbb{R}^n$. Recent work has tried to uncover the 'true' metric on the point cloud using distance-to-measure functions, in order to obtain more meaningful persistent homology results. Here we propose an alternative look at this problem: we posit that information on the point cloud is lost when restricting persistent homology to a single (correct) distance function. Instead, we show how by varying the distance function on the underlying space and analysing the corresponding shifts in the persistence diagrams, we can extract additional topological and geometrical information. Finally, we numerically show that non-isotropic persistent homology can extract information on orientation, orientational variance, and scaling of randomly generated point clouds with good accuracy and conduct some experiments on real-world data.