In this paper, we introduce a new family of descriptors for persistence diagrams. Our approach transforms these diagrams into elements of a finite-dimensional vector space using functionals based on the discrete measures they induce. While our focus is primarily on identity and frequency-based transformations, we do not restrict our approach exclusively to this types of techniques. We term this family of transformations as LITE (Lattice Integrated Topological Embedding) 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 an innovative perspective for data scientists but also critiques the current trajectory of literature on methodologies for vectorizing diagrams. It establishes a foundation for future progress in applying persistence diagrams to data analysis and machine learning under a more simple and effective lens.

翻译：本文提出了一类新的持续图描述符。该方法利用基于持续图所诱导离散测度的泛函，将这些图转化为有限维向量空间中的元素。尽管我们主要关注基于恒等式和频率的变换，但并未将方法局限于此类技术。我们将这类变换称为LITE（格集成拓扑嵌入），并证明该家族中某些成员在1-Kantorovitch-Rubinstein度量下具有稳定性，从而确保其对细微数据变化的敏感性。广泛的对比分析表明，我们的描述符与拓扑数据分析文献中的当前最先进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提供了创新视角，还对当前图示向量化方法学文献的发展方向进行了批判性审视，为未来在更简单有效的框架下将持续图应用于数据分析和机器学习奠定了坚实基础。