持久图空间的度量几何。

Mauricio Che, Fernando Galaz-García, Luis Guijarro, Ingrid Amaranta Membrillo Solis
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引用次数: 0

摘要

持久图是拓扑数据分析中的核心对象。在本文中,我们将研究持久图空间的局部和全局几何特性。为此,我们构建了一系列函数 D p , 1 ≤ p ≤ ∞ , 为每个度量对 (X, A) 分配一个尖度量空间 D p ( X , A ) 。此外,我们证明 D ∞ 在度量对的格罗莫夫-豪斯多夫收敛性方面是连续的,并证明 D p 保留了几个有用的度量特性,如对于 p∈ [ 1 , ∞ ) 的完备性和可分性,以及大地性和非大地性。 的情况下,D p 保留了几个有用的度量特性,如完整性和可分性;在 p = 2 的情况下,D p 保留了大地性和亚历山德罗夫意义上的非负曲率。对于后一种情况,我们描述了空图处方向空间的度量。我们还证明了在 D p ( X , A ) 上的博尔概率度量的弗雷谢特均值集,1 ≤ p ≤ ∞,具有有限第二矩和紧凑支持,是非空的。作为几何框架的一个应用,我们证明了欧氏持久图空间 D p ( R 2 n , Δ n ) , 1 ≤ n 且 1 ≤ p ∞ 具有无限覆盖维、豪斯多夫维、渐近维、阿苏阿德维和阿苏阿德-纳加塔维。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Metric geometry of spaces of persistence diagrams.

Metric geometry of spaces of persistence diagrams.

Metric geometry of spaces of persistence diagrams.

Metric geometry of spaces of persistence diagrams.

Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors D p , 1 p , that assign, to each metric pair (XA), a pointed metric space D p ( X , A ) . Moreover, we show that D is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that D p preserves several useful metric properties, such as completeness and separability, for p [ 1 , ) , and geodesicity and non-negative curvature in the sense of Alexandrov, for p = 2 . For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on D p ( X , A ) , 1 p , with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, D p ( R 2 n , Δ n ) , 1 n and 1 p < , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.

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