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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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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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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