Decorated merge trees for persistent topology
Decorated merge trees for persistent topology
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DOI：
10.1007/s41468-022-00089-3
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发表时间：
2021-03
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影响因子：
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通讯作者：
J. Curry;Haibin Hang;W. Mio;Tom Needham;Osman Berat Okutan
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文献类型：
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作者：
J. Curry;Haibin Hang;W. Mio;Tom Needham;Osman Berat Okutan
本文介绍了装饰合并树（dmt）作为一种新的持久化空间不变量。dmt将两个信息组合到一个单一的数据结构中，该数据结构可以区分合并树和持久同源不能单独区分的过滤。dmt的三种变体分别强调范畴论、表示论和持久性条形码，它们在理论和计算方面具有不同的优势。定义了DMTs的两个距离概念——交错距离和瓶颈距离，并证明了一个稳定性结果的层次结构，该层次结构改进和推广了现有的稳定性结果。为了克服这些距离固有的一些计算复杂性，我们提供了一种新的使用Gromov-Wasserstein耦合来计算可跟踪估计的交错距离的组合版本的最优合并树对齐。我们介绍了用于生成、可视化和比较由合成数据和真实数据派生的装饰合并树的计算框架。示例应用包括点云的比较，时间序列滑动窗口嵌入的持续同源性解释，分割脑肿瘤图像的拓扑特征可视化以及拓扑驱动的图对齐。
This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces. DMTs combine bothandinformation into a single data structure that distinguishes filtrations that merge trees and persistent homology cannot distinguish alone. Three variants on DMTs, which emphasize category theory, representation theory and persistence barcodes, respectively, offer different advantages in terms of theory and computation. Two notions of distance—an interleaving distance and bottleneck distance—for DMTs are defined and a hierarchy of stability results that both refine and generalize existing stability results is proved here. To overcome some of the computational complexity inherent in these distances, we provide a novel use of Gromov-Wasserstein couplings to compute optimal merge tree alignments for a combinatorial version of our interleaving distance which can be tractably estimated. We introduce computational frameworks for generating, visualizing and comparing decorated merge trees derived from synthetic and real data. Example applications include comparison of point clouds, interpretation of persistent homology of sliding window embeddings of time series, visualization of topological features in segmented brain tumor images and topology-driven graph alignment.