关于Lipschitz持久图矢量化的谱合成
On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations
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中文总结 AI 辅助
研究持久图矢量化，证明其Lipschitz扩展与特定上循环的等距同构，给出相关条件下上循环生成可合成簇的结论，还将谱合成扩展到可分度量对，探讨文献中例子是否有带谱合成的Lipschitz扩展。
中文摘要 AI 辅助
持久图将同调类的生灭表示为区间的多重集，持久图矢量化是从度量对\((X,A)\)上的持久图到巴拿赫空间\(E\)中值的映射\(D(X,A) \to E\)。我们证明了在\(D(X,A)\)的格罗滕迪克完备化\(K(X,A)\)上，矢量化的Lipschitz扩展（模常数）与\(K(X,A)\)在\(\ell^\infty(K(X,A),E)\)上的平移作用的有界\(1 -\)上循环之间的等距同构。接着证明若\(E\)上每个有界线性泛函将矢量化的Lipschitz扩展映射为从\(K(X,A)\)到\(\mathbb{C}\)的加法同态与\(\widehat{K(X,A)}\)上有限复正则博雷尔测度的傅里叶 - 斯蒂尔杰斯变换之和，则相关上循环生成可合成的簇。我们将谱合成扩展到可分度量对中持久图的Lipschitz矢量化，并展示文献中持久图矢量化的例子是否具有带谱合成的Lipschitz扩展。
英文摘要
Persistence diagrams are fundamental descriptors in topological data analysis. Many statistical and machine learning methods require persistence diagrams to be mapped into vector spaces or compared through kernels. Although many persistence diagram vectorizations and persistence kernels have been proposed, existing methods are typically developed as individual constructions, and existing work does not provide a canonical persistence diagram vectorization from which many existing vectorizations and kernels can be derived. In this paper, we develop such a canonical persistence diagram vectorization for certain classes of persistence diagrams using Fourier analysis on groups of virtual persistence diagrams. Spectral synthesis asks whether other persistence diagram vectorizations can be represented through this canonical vectorization. We prove this result for a class of Lipschitz persistence diagram vectorizations. We extend this result from uniformly discrete metric pairs to separable metric pairs.
