Persistence Flammatrix：用于核密度探索的多尺度持久同源
Persistence Flamelets: multiscale Persistent Homology for kernel density exploration
摘要：
近年来，人们对“数据形态”的研究产生了明显的兴趣。在定义“形状”的许多方法中，拓扑是最通用的一种，因为它根据对象的连接结构（连接组件）来描述对象（尺寸0的拓扑特征），循环存在越来越多的技术，通常被表示为拓扑数据分析，旨在估计固定对象的拓扑不变量;然而，当我们允许这个物体发生变化时，很少有人研究它的拓扑结构的演变。在这项工作中，我们定义的持久性flamestor，一个多尺度版本的TDA中最流行的工具之一，持久性景观。我们研究了它的理论特性，并展示了如何使用它来获得对KDE带宽参数的见解。
英文摘要：
In recent years there has been noticeable interest in the study of the "shape of data". Among the many ways a "shape" could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure: connected components (topological features of dimension 0), cycles (features of dimension 1) and so on. There is a growing number of techniques, generally denoted as Topological Data Analysis, aimed at estimating topological invariants of a fixed object; when we allow this object to change, however, little has been done to investigate the evolution in its topology. In this work we define the Persistence Flamelets, a multiscale version of one of the most popular tool in TDA, the Persistence Landscape. We examine its theoretical properties and we show how it could be used to gain insights on KDEs bandwidth parameter.
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