Convergence rates for persistence diagram estimation in Topological Data Analysis

Frédéric Chazal, Marc Glisse, Catherine Labruère, Bertrand Michel
Proceedings of the 31st International Conference on Machine Learning, PMLR 32(1):163-171, 2014.

Abstract

Computational topology has recently seen an important development toward data analysis, giving birth to Topological Data Analysis. Persistent homology appears as a fundamental tool in this field. We show that the use of persistent homology can be naturally considered in general statistical frameworks. We establish convergence rates of persistence diagrams associated to data randomly sampled from any compact metric space to a well defined limit diagram encoding the topological features of the support of the measure from which the data have been sampled. Our approach relies on a recent and deep stability result for persistence that allows to relate our problem to support estimation problems (with respect to the Gromov-Hausdorff distance). Some numerical experiments are performed in various contexts to illustrate our results.

Cite this Paper

BibTeX
@InProceedings{pmlr-v32-chazal14, title = {Convergence rates for persistence diagram estimation in Topological Data Analysis}, author = {Chazal, Frédéric and Glisse, Marc and Labruère, Catherine and Michel, Bertrand}, booktitle = {Proceedings of the 31st International Conference on Machine Learning}, pages = {163--171}, year = {2014}, editor = {Xing, Eric P. and Jebara, Tony}, volume = {32}, number = {1}, series = {Proceedings of Machine Learning Research}, address = {Bejing, China}, month = {22--24 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v32/chazal14.pdf}, url = {https://proceedings.mlr.press/v32/chazal14.html}, abstract = {Computational topology has recently seen an important development toward data analysis, giving birth to Topological Data Analysis. Persistent homology appears as a fundamental tool in this field. We show that the use of persistent homology can be naturally considered in general statistical frameworks. We establish convergence rates of persistence diagrams associated to data randomly sampled from any compact metric space to a well defined limit diagram encoding the topological features of the support of the measure from which the data have been sampled. Our approach relies on a recent and deep stability result for persistence that allows to relate our problem to support estimation problems (with respect to the Gromov-Hausdorff distance). Some numerical experiments are performed in various contexts to illustrate our results.} }
Endnote
%0 Conference Paper %T Convergence rates for persistence diagram estimation in Topological Data Analysis %A Frédéric Chazal %A Marc Glisse %A Catherine Labruère %A Bertrand Michel %B Proceedings of the 31st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2014 %E Eric P. Xing %E Tony Jebara %F pmlr-v32-chazal14 %I PMLR %P 163--171 %U https://proceedings.mlr.press/v32/chazal14.html %V 32 %N 1 %X Computational topology has recently seen an important development toward data analysis, giving birth to Topological Data Analysis. Persistent homology appears as a fundamental tool in this field. We show that the use of persistent homology can be naturally considered in general statistical frameworks. We establish convergence rates of persistence diagrams associated to data randomly sampled from any compact metric space to a well defined limit diagram encoding the topological features of the support of the measure from which the data have been sampled. Our approach relies on a recent and deep stability result for persistence that allows to relate our problem to support estimation problems (with respect to the Gromov-Hausdorff distance). Some numerical experiments are performed in various contexts to illustrate our results.
RIS
TY - CPAPER TI - Convergence rates for persistence diagram estimation in Topological Data Analysis AU - Frédéric Chazal AU - Marc Glisse AU - Catherine Labruère AU - Bertrand Michel BT - Proceedings of the 31st International Conference on Machine Learning DA - 2014/01/27 ED - Eric P. Xing ED - Tony Jebara ID - pmlr-v32-chazal14 PB - PMLR DP - Proceedings of Machine Learning Research VL - 32 IS - 1 SP - 163 EP - 171 L1 - http://proceedings.mlr.press/v32/chazal14.pdf UR - https://proceedings.mlr.press/v32/chazal14.html AB - Computational topology has recently seen an important development toward data analysis, giving birth to Topological Data Analysis. Persistent homology appears as a fundamental tool in this field. We show that the use of persistent homology can be naturally considered in general statistical frameworks. We establish convergence rates of persistence diagrams associated to data randomly sampled from any compact metric space to a well defined limit diagram encoding the topological features of the support of the measure from which the data have been sampled. Our approach relies on a recent and deep stability result for persistence that allows to relate our problem to support estimation problems (with respect to the Gromov-Hausdorff distance). Some numerical experiments are performed in various contexts to illustrate our results. ER -
APA
Chazal, F., Glisse, M., Labruère, C. & Michel, B.. (2014). Convergence rates for persistence diagram estimation in Topological Data Analysis. Proceedings of the 31st International Conference on Machine Learning, in Proceedings of Machine Learning Research 32(1):163-171 Available from https://proceedings.mlr.press/v32/chazal14.html.

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