Metric geometry of spaces of persistence diagrams

Che, Mauricio; Galaz Garcia, Fernando; Guijarro, Luis; Membrillo Solis, Ingrid

Abstract

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 Dp, 1≤p≤∞, that assign, to each metric pair (X, A), a pointed metric space Dp(X, A). Moreover, we show that D∞ is sequentially continuous with respect to the Gromov–Hausdorff convergence of metric pairs, and we prove that Dp 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 Dp(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, Dp(R2n, Δn), 1≤n and 1≤p<∞, has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad–Nagata dimensions.

Citation

Che, M., Galaz Garcia, F., Guijarro, L., & Membrillo Solis, I. (2024). Metric geometry of spaces of persistence diagrams. Journal of Applied and Computational Topology, 8(8), 2197-2246. https://doi.org/10.1007/s41468-024-00189-2

• Journal Article Type: Article
• Acceptance Date: Jul 12, 2024
• Online Publication Date: Sep 3, 2024
• Publication Date: Dec 1, 2024
• Deposit Date: Jul 25, 2024
• Publicly Available Date: Sep 6, 2024
• Journal: Journal of Applied and Computational Topology
• Print ISSN: 2367-1726
• Electronic ISSN: 2367-1734
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 8
• Issue: 8
• Pages: 2197-2246
• DOI: https://doi.org/10.1007/s41468-024-00189-2
• Keywords: 54F45, 53C23, Asymptotic dimension, Gromov–Hausdorff convergence, Persistence diagram, Alexandrov spaces, 55N31, Fré chet mean set, Metric pairs, 64R40
• Public URL: https://durham-repository.worktribe.com/output/2612996
• Other Repo URL: https://arxiv.org/abs/2109.14697