首次证明单个单纯形插入后谱拓扑特征的稳定性边界。
Lipschitz Bounds for Persistent Laplacian Eigenvalues under One-Simplex Insertions
- 建立上持久拉普拉斯算子在添加单个单纯形时的统一利普希茨界
- 任意特征值变化不超过该单纯形边界的欧氏范数的两倍
- 为动态数据中的谱拓扑分析提供可信赖的误差控制依据
持久拉普拉斯算子用于追踪数据在不同尺度下的形状与结构变化，广泛应用于生物学、物理学和机器学习。其特征值是滤链中几何与拓扑特征的简洁描述。尽管已有研究建立了这些算子的全局代数稳定性，但添加一个单纯形（如顶点、边或三角形）后单个特征值的精确变化仍未知。这至关重要，因为下游工具如热核签名和谱神经网络直接依赖于这些特征值。本文首次证明：在插入一个单纯形后，所有上持久拉普拉斯特征值的变化量不超过该单纯形边界欧氏范数的两倍，且与滤链尺度和复形大小无关。这一结果首次为谱拓扑数据分析提供了特征值级别的鲁棒性保证，确保谱特征在局部更新下保持稳定，并支持动态数据场景中的可靠误差控制。
原文摘要 · Abstract (English)
Persistent Laplacians are matrix operators that track how the shape and structure of data transform across scales and are popularly adopted in biology, physics, and machine learning. Their eigenvalues are concise descriptors of geometric and topological features in a filtration. Although earlier work established global algebraic stability for these operators, the precise change in a single eigenvalue when one simplex, such as a vertex, edge, or triangle, is added has remained unknown. This is important because downstream tools, including heat-kernel signatures and spectral neural networks, depend directly on these eigenvalues. We close this gap by proving a uniform Lipschitz bound: after inserting one simplex, every up-persistent Laplacian eigenvalue can vary by at most twice the Euclidean norm of that simplex's boundary, independent of filtration scale and complex size. This result delivers the first eigenvalue-level robustness guarantee for spectral topological data analysis. It guarantees that spectral features remain stable under local updates and enables reliable error control in dynamic data settings.
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