Abstract Persistence barcode is a topological summary for persistent homology to exhibit topological features with different persistence. Persistence rank function (PRF), derived from persistence barcode, organizes persistence Betti numbers in the form of an integer-valued function. To obtain topological patterns of objects such as point clouds represented by finite-dimensional vectors for machine learning classification tasks, the vectorizing representations of barcodes is generated via decomposing PRF on a system of Haar basis. Theoretically, the generated vectorizing representation is proved to have 1-Wasserstein stability. In practice, to reduce training time and achieve better results, a technique of dimensionality reduction through out-of-sample mapping in supervised manifold learning is used to generate a low-dimensional vector. Experiments demonstrate that the representation is effective for capturing the topological patterns of data sets. Moreover, the classification of porous structures has become an essential problem in the fields such as material science in recent decades. The proposed method is successfully applied to distinguish porous structures on a novel data set of porous models.

中文翻译：

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持久性条形码矢量化在多孔结构模式分类中的应用

摘要 持久性条码是持久同源性的拓扑概括，表现出具有不同持久性的拓扑特征。持久性等级函数 (PRF) 源自持久性条形码，以整数值函数的形式组织持久性 Betti 数。为了获得用于机器学习分类任务的有限维向量表示的对象（例如点云）的拓扑模式，通过在 Haar 基础上分解 PRF 生成条形码的向量化表示。理论上，生成的向量化表示被证明具有 1-Wasserstein 稳定性。在实践中，为了减少训练时间并获得更好的结果，使用监督流形学习中通过样本外映射进行降维的技术来生成低维向量。实验表明，该表示对于捕获数据集的拓扑模式是有效的。此外，多孔结构的分类已成为近几十年来材料科学等领域的重要问题。所提出的方法已成功应用于区分多孔模型新数据集上的多孔结构。