In this paper, we present a novel family of descriptors for persistence diagrams, reconceptualizing them as signals in $\mathbb R^2_+$. This marks a significant advancement in Topological Data Analysis. Our methodology transforms persistence diagrams into a finite-dimensional vector space through functionals of the discrete measures induced by these diagrams. While our focus is primarily on frequency-based transformations, we do not restrict our approach exclusively to this types of techniques. We term this family of transformations as $Persistence$ $Signals$ and prove stability for some members of this family against the 1-$Kantorovitch$-$Rubinstein$ metric, ensuring its responsiveness to subtle data variations. Extensive comparative analysis reveals that our descriptor performs competitively with the current state-of-art from the topological data analysis literature, and often surpasses, the existing methods. This research not only introduces a groundbreaking perspective for data scientists but also establishes a foundation for future innovations in applying persistence diagrams in data analysis and machine learning.

翻译：本文提出了一类新颖的持久图描述子，将其重新概念化为$\mathbb R^2_+$空间中的信号。这标志着拓扑数据分析领域的重要进展。我们的方法通过持久图所诱导的离散测度的泛函，将其转化为有限维向量空间。虽然我们主要关注基于频率的变换，但并不将该方法局限于此类技术。我们将这一系列变换命名为"持续信号"，并证明了该系列中某些成员在1-康托罗维奇-鲁宾斯坦度量下的稳定性，确保其对微妙数据变化的敏感性。广泛的对比分析表明，我们的描述子与当前拓扑数据分析文献中的最新方法相比具有竞争力，且常常超越现有方法。本研究不仅为数据科学家引入了一种开创性视角，也为未来在数据分析和机器学习中应用持久图的创新奠定了坚实基础。