标题： 使用多尺度高斯链接积分的结点数据分析 显示英文标题

标题： Knot data analysis using multiscale Gauss link integral

摘要： 在过去十年中，拓扑数据分析（TDA）已成为数据科学中的一个强大工具。TDA的主要技术是持久同调，它使用代数拓扑通过点云数据的过滤来追踪拓扑不变量。尽管结理论及相关主题是数学研究的重点，但由于缺乏定位和量化，它们在实际应用中的成功案例相当有限。我们通过引入结数据分析（KDA），一种新的范式，将曲线分割和多尺度分析融入高斯链环积分中，解决了这些挑战。由此产生的多尺度高斯链环积分（mGLI）在适当的尺度上恢复了结和链环的整体拓扑属性，但提供了多尺度特征向量以捕捉各个曲线段在不同尺度上的局部结构和连通性。所提出的mGLI在基准蛋白质灵活性分析中显著优于其他最先进的方法，包括早期基于持久同调的方法。我们的方法使得人工智能（AI）和KDA能够整合到一般曲线类对象和数据中。 显示英文摘要

摘要： In the past decade, topological data analysis (TDA) has emerged as a powerful approach in data science. The main technique in TDA is persistent homology, which tracks topological invariants over the filtration of point cloud data using algebraic topology. Although knot theory and related subjects are a focus of study in mathematics, their success in practical applications is quite limited due to the lack of localization and quantization. We address these challenges by introducing knot data analysis (KDA), a new paradigm that incorporating curve segmentation and multiscale analysis into the Gauss link integral. The resulting multiscale Gauss link integral (mGLI) recovers the global topological properties of knots and links at an appropriate scale but offers multiscale feature vectors to capture the local structures and connectivities of each curve segment at various scales. The proposed mGLI significantly outperforms other state-of-the-art methods in benchmark protein flexibility analysis, including earlier persistent homology-based methods. Our approach enables the integration of artificial intelligence (AI) and KDA for general curve-like objects and data.