Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to be challenging as the space lacks an inner product. For that reason, transforming these diagrams in a way that is compatible with machine learning is an important topic currently researched in TDA. In this paper, our main contribution consists of three components. First, we develop a general and unifying framework of vectorizing diagrams that we call the \textit{Persistence Curves} (PCs), and show that several well-known summaries, such as Persistence Landscapes, fall under the PC framework. Second, we propose several new summaries based on PC framework and provide a theoretical foundation for their stability analysis. Finally, we apply proposed PCs to two applications---texture classification and determining the parameters of a discrete dynamical system; their performances are competitive with other TDA methods.

翻译：持久性图表是地形数据分析领域的主要工具之一。 它们包含关于数据形状的富有成果的信息。 在持久性图表空间上使用机器学习算法证明具有挑战性,因为空间缺乏内部产品。 因此,将这些图表转换成与机器学习兼容的方式是目前TDA研究的一个重要专题。 在本文中,我们的主要贡献包括三个组成部分。 首先,我们开发了一个通缩图的一般和统一框架,我们称之为\ textit{Persistence Curves}(PCs), 并显示一些众所周知的概要, 如Persistence Landscaps, 属于PC框架的范畴。 其次,我们根据PC框架提出了几个新的摘要,为它们的稳定性分析提供了理论基础。 最后,我们将个人计算机应用于两种应用-文字分类并确定离散动态系统的参数;它们的性能与其他TDA方法相比是竞争性的。