基于持久上同调的密度鲁棒球坐标
Density-Robust Spherical Coordinates from Persistent Cohomology
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中文总结 AI 辅助
研究如何从持久上同调构造密度鲁棒球坐标，核心方法是扩展圆坐标子采样和对齐框架到\(S^2\)，将对齐问题表述为球Procrustes问题并建立松弛保证，贡献是对非均匀采样鲁棒，避免经典方法内存瓶颈，实验验证了其有效性和可扩展性。
中文摘要 AI 辅助
持久上同调为构建反映数据拓扑结构的非线性坐标提供了一个原理框架。然而，这些拓扑坐标会因采样密度不均匀而严重扭曲，限制了其在实际数据中的应用。虽然最近已开发出密度鲁棒的圆坐标，但向球坐标的扩展仍是一个开放挑战。本文介绍了第一种基于持久上同调的密度鲁棒球坐标构造方法。我们将圆坐标的子采样和对齐框架扩展到\(S^2\)，先在通过拒绝采样获得的近似均匀子样本上计算球坐标，再将它们组合成全局一致映射。主要数学难点是对齐独立计算的球值坐标，我们将此挑战表述为球Procrustes问题并为可计算的欧几里得松弛建立近似保证。实验表明该方法在严重采样偏差下能准确恢复坐标且可扩展到10000点的数据集。
英文摘要
Persistent cohomology provides a principled framework for constructing nonlinear coordinates that reflect the topology of data. However, these topological coordinates can be severely distorted by non-uniform sampling density, limiting their applicability to real-world data. While density-robust circular coordinates have recently been developed, the extension to spherical coordinates remains an open challenge: unlike the circular case, spherical coordinates are obtained through a nonlinear variational problem for sphere-valued maps, to which existing density-correction mechanisms are not directly applicable. In this paper, we introduce the first density-robust construction of spherical coordinates from persistent cohomology. Rather than modifying the coordinate optimization itself, we extend a subsampling-and-alignment framework for circular coordinates to $S^2$, which first computes spherical coordinates on approximately uniform subsamples obtained by rejection sampling and then combines them into a global consensus map. The principal mathematical difficulty is the alignment of independently computed sphere-valued coordinates. We formulate this challenge as a spherical Procrustes problem and establish approximation guarantees for a computationally tractable Euclidean relaxation. Our resulting construction is robust to non-uniform sampling and retains the accuracy of classical spherical coordinates under uniform sampling. Moreover, by computing persistent cohomology only on fixed-size subsamples, our approach avoids the quartic memory bottleneck of the classical spherical coordinate pipeline and scales to substantially larger datasets. We conduct experiments on synthetic data and demonstrate accurate coordinate recovery under severe sampling bias and scalability to datasets of 10,000 points.
