结构化数据中多尺度循环的超图方法
Hypergraphs for multiscale cycles in structured data
AI总结：
本文提出 hyperTDA 拓扑流程，将持续同调生成元转化为加权超图，统一局部与全局度量，在噪声下鲁棒地分析多尺度空间结构，并成功应用于动物运动轨迹分类与金融数据。
摘要：
在现代物理、工程、生命和社会科学中，科学数据的规模和复杂性都在不断增长。例如，空间结构是许多最重要的现实世界复杂系统的标志，但对其分析充满统计挑战。拓扑数据分析可以为复杂系统提供强大的计算窗口。在此，我们提出了一个框架，用于扩展和解释持续同调摘要，以在多个尺度上分析空间数据。我们引入了 hyperTDA，这是一种拓扑流程，它统一了局部（如测地线）和全局（如欧几里得）度量，同时不丢失空间信息，即使在存在噪声的情况下也是如此。同调生成元为空间结构提供了一种优雅且灵活的描述，并能够以可解释的方式捕获持续同调计算出的信息。在此，持续同调计算出的信息被转换为加权超图，其中超边对应于同调生成元。我们考虑了不同的生成元选择（例如拟阵生成元或最小生成元），并发现中心性和社区检测对任一选择都具有鲁棒性。我们将 hyperTDA 与现有的几何度量进行比较，并验证了其对噪声的鲁棒性。我们展示了在生态学、生物物理学和生物学中经常出现的空间曲线以及高维金融数据集上计算高阶拓扑结构的能力。我们发现 hyperTDA 能够区分来自具有里程碑意义的 2020 年 AnDi 挑战赛的合成轨迹，并在数据有限的情况下量化不同动物物种的运动。
英文摘要：
Scientific data has been growing in both size and complexity across the modern physical, engineering, life and social sciences. Spatial structure, for example, is a hallmark of many of the most important real-world complex systems, but its analysis is fraught with statistical challenges. Topological data analysis can provide a powerful computational window on complex systems. Here we present a framework to extend and interpret persistent homology summaries to analyse spatial data across multiple scales. We introduce hyperTDA, a topological pipeline that unifies local (e.g. geodesic) and global (e.g. Euclidean) metrics without losing spatial information, even in the presence of noise. Homology generators offer an elegant and flexible description of spatial structures and can capture the information computed by persistent homology in an interpretable way. Here the information computed by persistent homology is transformed into a weighted hypergraph, where hyperedges correspond to homology generators. We consider different choices of generators (e.g. matroid or minimal) and find that centrality and community detection are robust to either choice. We compare hyperTDA to existing geometric measures and validate its robustness to noise. We demonstrate the power of computing higher-order topological structures on spatial curves arising frequently in ecology, biophysics, and biology, but also in high-dimensional financial datasets. We find that hyperTDA can select between synthetic trajectories from the landmark 2020 AnDi challenge and quantifies movements of different animal species, even when data is limited.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
