提出一种能精准保持拓扑距离的新型图谱表示方法。
Persistence Spheres: Bi-continuous Representations of Persistence Diagrams
- 将图谱映射到线性空间，同时保持距离与逆映射连续
- 在多种数据上表现优于或媲美现有方法
- 适合需要精确拓扑几何的应用场景
我们提出持久性球（persistence spheres），一种全新的持久性图谱函数表示方法。与现有的嵌入方式（如持久性图像、景观或核方法）不同，持久性球提供双连续映射：在1-Wasserstein距离下是Lipschitz连续的，并且在其像集上存在连续逆映射。这在理论上最优地保证了稳定性与几何保真度，使持久性球成为最贴近图谱的Wasserstein几何的线性空间表示。我们推导出持久性球的显式公式，表明其可高效计算并实现最小开销的并行化。我们在涉及函数数据、时间序列、图、网格和点云的多样化回归与分类任务中进行实证评估。在这些基准测试中，持久性球始终表现达到或接近当前最优水平，优于或媲美持久性图像、持久性景观及切片Wasserstein核。
原文摘要 · Abstract (English)
We introduce persistence spheres, a novel functional representation of persistence diagrams. Unlike existing embeddings (such as persistence images, landscapes, or kernel methods), persistence spheres provide a bi-continuous mapping: they are Lipschitz continuous with respect to the 1-Wasserstein distance and admit a continuous inverse on their image. This ensures, in a theoretically optimal way, both stability and geometric fidelity, making persistence spheres the representation that most closely mirrors the Wasserstein geometry of PDs in linear space. We derive explicit formulas for persistence spheres, showing that they can be computed efficiently and parallelized with minimal overhead. Empirically, we evaluate them on diverse regression and classification tasks involving functional data, time series, graphs, meshes, and point clouds. Across these benchmarks, persistence spheres consistently deliver state-of-the-art or competitive performance compared to persistence images, persistence landscapes, and the sliced Wasserstein kernel.
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