One of the primary areas of interest in applied algebraic topology is
persistent homology, and, more specifically, the persistence diagram.
Persistence diagrams have also become objects of interest in topological data
analysis. However, persistence diagrams do not naturally lend themselves to
statistical goals, such as inferring certain population characteristics,
because their complicated structure makes common algebraic operations--such as
addition, division, and multiplication-- challenging (e.g., the mean might not
be unique). To bypass these issues, several functional summaries of persistence
diagrams have been proposed in the literature (e.g. landscape and silhouette
functions). The problem of analyzing a set of persistence diagrams then becomes
the problem of analyzing a set of functions, which is a topic that has been
studied for decades in statistics. First, we review the various functional
summaries in the literature and propose a unified framework for the functional
summaries. Then, we generalize the definition of persistence landscape
functions, establish several theoretical properties of the persistence
functional summaries, and demonstrate and discuss their performance in the
context of classification using simulated prostate cancer histology data, and
two-sample hypothesis tests comparing human and monkey fibrin images, after
developing a simulation study using a new data generator we call the Pickup
Sticks Simulator (STIX).

本研究提出一种在应用代数拓扑中的持久同调领域中进行分类研究的方法，即通过对持久图的函数汇总，建立持久景观函数的定义、理论性质及性能进行理论与实践分析，结果表明其分类效果优越。