A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams
A Riemannian Framework for Statistical Analysis of Topological Persistence Diagrams
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拓扑持久图统计分析的黎曼框架
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
P. Turaga
中科院分区：
文献类型：
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作者：
Rushil Anirudh;Vinay Venkataraman;K. Ramamurthy;P. Turaga
拓扑数据分析正在成为研究高维特征空间的一种流行方法，无需任何上下文线索或假设。本文关注的是一个流行的拓扑特征，即数据集中d维洞的数量，也称为Betti-d数。Betti数在不同尺度上的持久性被编码成持久性图（PD），该图显示了这些孔随着尺度的变化而产生和死亡的时间。比较pd的常用方法是通过点对点匹配，这是由n-Wasserstein度量给出的。然而，这种方法的一个很大的缺点是需要在计算距离之前解决点之间的对应关系，对于n个点，复杂度按O（n3）增长。相反，我们建议使用建立在黎曼几何基础上的全新框架，将pd建模为二维概率密度函数，在希尔伯特球的平方根框架中表示。生成的空间使用用于常见操作的封闭形式表达式更加直观。距离度量是1)无对应性的，也2)独立于数据集中的点的数量。对于[0,1]2的K K离散化，pd之间计算距离的复杂性现在按照O（K2）增长。这也使得微分几何中现有的机器能够用于pd的统计分析，如计算平均值、测地线、分类等。我们报告了与Wasserstein度量的竞争结果，在更低的计算负载下，表明了所提出方法的有利特性。
Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of d–dimensional holes in the dataset, also known as the Betti–d number. The persistence of the Betti numbers over various scales is encoded into a persistence diagram (PD), which indicates the birth and death times of these holes as scale varies. A common way to compare PDs is by a pointto-point matching, which is given by the n-Wasserstein metric. However, a big drawback of this approach is the need to solve correspondence between points before computing the distance, for n points, the complexity grows according to O(n3). Instead, we propose to use an entirely new framework built on Riemannian geometry, that models PDs as 2D probability density functions that are represented in the square-root framework on a Hilbert Sphere. The resulting space is much more intuitive with closed form expressions for common operations. The distance metric is 1) correspondence-free and also 2) independent of the number of points in the dataset. The complexity of computing distance between PDs now grows according to O(K2), for a K K discretization of [0, 1]2. This also enables the use of existing machinery in differential geometry towards statistical analysis of PDs such as computing the mean, geodesics, classification etc. We report competitive results with the Wasserstein metric, at a much lower computational load, indicating the favorable properties of the proposed approach.
DOI：
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发表时间：
2015-07
期刊：
J. Mach. Learn. Res.
影响因子：
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作者：
Henry Adams;T. Emerson;M. Kirby;R. Neville;C. Peterson;Patrick D. Shipman;Sofya Chepushtanova;Eric M. Hanson;Francis C. Motta;Lori Ziegelmeier
通讯作者：
Henry Adams;T. Emerson;M. Kirby;R. Neville;C. Peterson;Patrick D. Shipman;Sofya Chepushtanova;Eric M. Hanson;Francis C. Motta;Lori Ziegelmeier