The DOI matches its Crossref record; 34 of the 35 references it cites were found in a registry or an index.
Probability measures on the space of persistence diagrams
Open accessAbstract
This paper shows that the space of persistence diagrams has properties that allow for the definition of probability measures which support expectations, variances, percentiles and conditional probabilities. This provides a theoretical basis for a statistical treatment of persistence diagrams, for example computing sample averages and sample variances of persistence diagrams. We first prove that the space of persistence diagrams with the Wasserstein metric is complete and separable. We then prove a simple criterion for compactness in this space. These facts allow us to show the existence of the standard statistical objects needed to extend the theory of topological persistence to a much larger set of applications.
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“Theorem 4.11 (Mileyko et al. 2011). The metric space ( ˆ B p , ˆ d p ) is complete and separable.”
From Topological pattern recognition for point cloud data (2014) · cites this paper as Mileyko et al. 2011
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“One obstacle to realizing this goal is the fact that the metric space ( B ∞ , d p ) is not complete.”
From Topological pattern recognition for point cloud data (2014) · cites this paper as Mileyko et al. (2011)
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“The resulting metric space is simpler to work with computationally, and all the existence results from Mileyko et al. (2011) are also shown to hold in Turner et al. (2014).”
From Topological pattern recognition for point cloud data (2014) · cites this paper as Mileyko et al. (2011)
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Cites
On Estimation of a Probability Density Function and Mode
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Flagged first, then by citations
1 of the 35 references couldn’t be matched to a known paper.
Research works printed but not linked: 7
- de Silva V, 2004, Symp. on Point-based Graphics, 157
- Edelsbrunner H, 2008, Discrete and Computational Geometry. Twenty Years Later, 257
- Wasserstein L N, 1969, Probl. Inf. Transm., 5, 47
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Cited by
Sorted by citations
Statistical Aspects of Wasserstein Distances
Panaretos, V. M., & Zemel, Y. (2018) · Annual Review of Statistics and Its Application · doi:10.1146/annurev-statistics-030718-104938 · Cited by 879
A roadmap for the computation of persistent homology
Otter, N., Porter, M. A., Tillmann, U., Grindrod, P., & Harrington, H. A. (2017) · EPJ Data Science · doi:10.1140/epjds/s13688-017-0109-5 · Cited by 769
Statistical topological data analysis using persistence landscapesPreprint
Bubenik, P. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1207.6437 · Cited by 628
Persistence Images: A Stable Vector Representation of Persistent HomologyPreprint
Adams, H., Chepushtanova, S., Emerson, T., Hanson, E., Kirby, M. J. L., Motta, F. C., Neville, R., Peterson, C., Shipman, P. D., & Ziegelmeier, L. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1507.06217 · Cited by 517
Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis
Perea, J. A., & Harer, J. (2015) · Foundations of Computational Mathematics · doi:10.1007/s10208-014-9206-z · Cited by 306
Self-citation
Metadata
- Authors
- Mileyko, Y., Mukherjee, S., & Harer, J.
- Affiliations
- Duke University (US)
- Venue
- Inverse Problems
- Publisher
- IOP Publishing
- Published
- 2011-11-10
- Volume
- 27
- Issue
- 12
- Pages
- 124007
- ISSN
- 0266-5611 (ISSN-L)
- 1361-6420
- OpenAlex
- W2037817770
- Cited by
- 249
- References
- 35
- Citations by year
- 2026: 15 · 2025: 19 · 2024: 24 · 2023: 20 · 2022: 19 · 2021: 39 · 2020: 26 · 2019: 13 · 2018: 11 · 2017: 9 · 2016: 13 · 2015: 14 · 2014: 13 · 2013: 7 · 2012: 6
- FWCI
- 8.15
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Inverse Problems (publishedVersion)
- Funders
- National Science Foundation (1049290)
- Princeton University
- Brazilian Biodivesity Fund
- National Institutes of Health
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- persistence diagrams, probability measures, Wasserstein distance, topological data analysis, compactness, sample variance, conditional probability
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