On the Bootstrap for Persistence Diagrams and Landscapes
https://doi.org/10.18255/1818-1015-2013-6-111-120
Abstract
Persistent homology probes topological properties from point clouds and functions. By looking at multiple scales simultaneously, one can record the births and deaths of topological features as the scale varies. In this paper we use a statistical technique, the empirical bootstrap, to separate topological signal from topological noise. In particular, we derive confidence sets for persistence diagrams and confi- dence bands for persistence landscapes.
The article is published in the author’s wording.
Supporting agencies: Sivaraman Balakrishnan
About the Authors
F. Chazal
Geometrica INRIA Saclay
Russian Federation
Russian Federation
B.T. Fasy
Tulane University
Russian Federation
Russian Federation
Computer Science Department,
Stanley Thomas 303 New Orleans, LA 70118
F. Lecci
Carnegie Mellon University
Russian Federation
Russian Federation
Department of Statistics,
Baker Hall 132 Pittsburgh, PA 15213
A. Rinaldo
Carnegie Mellon University
Russian Federation
Russian Federation
Department of Statistics,
Baker Hall 132 Pittsburgh, PA 15213
A. Singh
Carnegie Mellon University
Russian Federation
Russian Federation
Machine Learning Department,
Gates Hillman Centers, 8203 5000 Forbes Avenue Pittsburgh, PA 15213-3891
L. Wasserman
Carnegie Mellon University
Russian Federation
Russian Federation
Department of Statistics,
Baker Hall 132 Pittsburgh, PA 15213
References
1. Sivaraman Balakrishnan, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Aarti Singh, and Larry Wasserman. Statistical inference for persistent homology, 2013. arXiv:1303.7117.
2. Peter Bubenik. Statistical topology using persistence landscapes, 2012. arXiv:1207.6437.
3. Fr´ed´eric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The structure and stability of persistence modules, July 2012. arXiv:1207.3674.
4. David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete Comput. Geom., 37(1):103–120, 2007.
5. Anthony Christopher Davison and D. V. Hinkley. Bootstrap Methods and Their Application, volume 1. Cambridge UP, 1997.
6. Persi Diaconis, Susan Holmes, and Mehrdad Shahshahani. Sampling from a manifold, 2012. arXiv:1206.6913.
7. Herbert Edelsbrunner and John Harer. Computational Topology. An Introduction. Amer. Math. Soc., Providence, RI, 2010.
8. Bradley Efron. Bootstrap methods: Another look at the jackknife. Ann. Statist. , pages 1–26, 1979.
9. Bradley Efron, Robert Tibshirani, John D. Storey, and Virginia Tusher. Empirical Bayes analysis of a microarray experiment. J. Amer. Statist. Assoc., 96(456):1151–1160, 2001.
10. Evarist Gin´e and Armelle Guillou. Rates of strong uniform consistency for multivariate kernel density estimators. In Annales de l’Institut Henri Poincare (B) Probability and Statistics, volume 38, pages 907–921. Elsevier, 2002.
11. Evarist Gin´e and Joel Zinn. Bootstrapping general empirical measures. The Annals of Probability, pages 851–869, 1990.
12. Michael R. Kosorok. Introduction to Empirical Processes and Semiparametric Inference. Springer, 2008.
13. Aad Van der Vaart. Asymptotic statistics, volume 3. Cambridge university press, 2000.
14. Aad Van der Vaart and Jon Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, 1996.
Review
For citations:
Chazal F., Fasy B., Lecci F., Rinaldo A., Singh A., Wasserman L. On the Bootstrap for Persistence Diagrams and Landscapes. Modeling and Analysis of Information Systems. 2013;20(6):111-120. https://doi.org/10.18255/1818-1015-2013-6-111-120
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