Persistent Homology (PH) is a useful tool to study the underlying structure of a data set. Persistence Diagrams (PDs), which are 2D multisets of points, are a concise summary of the information found by studying the PH of a data set. However, PDs are difficult to incorporate into a typical machine learning workflow. To that end, two main methods for representing PDs have been developed: kernel methods and vectorization methods. In this paper we propose a new finite-dimensional vector, called the interconnectivity vector, representation of a PD adapted from Bag-of-Words (BoW). This new representation is constructed to demonstrate the connections between the homological features of a data set. This initial definition of the interconnectivity vector proves to be unstable, but we introduce a stabilized version of the vector and prove its stability with respect to small perturbations in the inputs. We evaluate both versions of the presented vectorization on several data sets and show their high discriminative power.

中文翻译：

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互连矢量：持久同源性的有限维矢量表示

持久同源性（PH）是研究数据集基础结构的有用工具。持久性图（PD）是点的2D多集，是通过研究数据集的PH找到的信息的简要摘要。但是，PD很难整合到典型的机器学习工作流程中。为此，已经开发了两种表示PD的主要方法：内核方法和矢量化方法。在本文中，我们提出了一种新的有限维矢量，称为互连矢量，它是一种基于词袋（BoW）的PD的表示。这种新的表示形式旨在证明数据集的同源性特征之间的联系。互连矢量的最初定义被证明是不稳定的，但是我们引入了向量的稳定版本，并证明了其对于输入中的小扰动的稳定性。我们在几个数据集上评估了提出的矢量化的两个版本，并显示了它们的高判别力。