Shen Li, Feng Hongsong, Li Fengling, Lei Fengchun, Wu Jie, Wei Guo-Wei
Department of Mathematics, Michigan State University, East Lansing, MI 48824.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China.
Proc Natl Acad Sci U S A. 2024 Oct 15;121(42):e2408431121. doi: 10.1073/pnas.2408431121. Epub 2024 Oct 11.
In the past decade, topological data analysis has emerged as a powerful algebraic topology approach in data science. 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 paradigm that incorporates 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 and offers a multiscale geometric topology approach to capture the local structures and connectivities in data. By integration with machine learning or deep learning, the proposed mGLI significantly outperforms other state-of-the-art methods across various benchmark problems in 13 intricately complex biological datasets, including protein flexibility analysis, protein-ligand interactions, human Ether-à-go-go-Related Gene potassium channel blockade screening, and quantitative toxicity assessment. Our KDA opens a research area-knot deep learning-in data science.
在过去十年中，拓扑数据分析已成为数据科学中一种强大的代数拓扑方法。尽管纽结理论及相关学科是数学研究的重点，但由于缺乏局部化和量化，它们在实际应用中的成功相当有限。我们通过引入纽结数据分析（KDA）来应对这些挑战，这是一种将曲线分割和多尺度分析纳入高斯链积分的范式。由此产生的多尺度高斯链积分（mGLI）在适当的尺度上恢复了纽结和链环的全局拓扑性质，并提供了一种多尺度几何拓扑方法来捕捉数据中的局部结构和连通性。通过与机器学习或深度学习相结合，所提出的mGLI在13个极其复杂的生物数据集中的各种基准问题上显著优于其他现有方法，包括蛋白质柔性分析、蛋白质-配体相互作用、人类醚-à-去-去相关基因钾通道阻断筛选和定量毒性评估。我们的KDA在数据科学中开辟了一个研究领域——纽结深度学习。
