RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds

Thibault de Surrel, Felix Hensel, Mathieu Carrière, Théo Lacombe, Yuichi Ike, Hiroaki Kurihara, Marc Glisse, Frédéric Chazal
Proceedings of Topological, Algebraic, and Geometric Learning Workshops 2022, PMLR 196:96-106, 2022.

Abstract

The use of topological descriptors in modern machine learning applications, such as persistence diagrams (PDs) arising from Topological Data Analysis (TDA), has shown great potential in various domains. However, their practical use in applications is often hindered by two major limitations: the computational complexity required to compute such descriptors exactly, and their sensitivity to even low-level proportions of outliers. In this work, we propose to bypass these two burdens in a data-driven setting by entrusting the estimation of (vectorization of) PDs built on top of point clouds to a neural network architecture that we call RipsNet. Once trained on a given data set, RipsNet can estimate topological descriptors on test data very efficiently with generalization capacity. Furthermore, we prove that RipsNet is robust to input perturbations in terms of the 1-Wasserstein distance, a major improvement over the standard computation of PDs that only enjoys Hausdorff stability, yielding RipsNet to substantially outperform exactly-computed PDs in noisy settings. We showcase the use of RipsNet on both synthetic and real-world data. Our implementation will be made freely and publicly available as part of the open-source library Gudhi.

Cite this Paper

BibTeX
@InProceedings{pmlr-v196-surrel22a, title = {RipsNet: A General Architecture for Fast and Robust Estimation of the Persistent Homology of Point Clouds}, author = {de Surrel, Thibault and Hensel, Felix and Carri\`{e}re, Mathieu and Lacombe, Th\'{e}o and Ike, Yuichi and Kurihara, Hiroaki and Glisse, Marc and Chazal, Fr\'{e}d\'{e}ric}, booktitle = {Proceedings of Topological, Algebraic, and Geometric Learning Workshops 2022}, pages = {96--106}, year = {2022}, editor = {Cloninger, Alexander and Doster, Timothy and Emerson, Tegan and Kaul, Manohar and Ktena, Ira and Kvinge, Henry and Miolane, Nina and Rieck, Bastian and Tymochko, Sarah and Wolf, Guy}, volume = {196}, series = {Proceedings of Machine Learning Research}, month = {25 Feb--22 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v196/surrel22a/surrel22a.pdf}, url = {https://proceedings.mlr.press/v196/surrel22a.html}, abstract = {The use of topological descriptors in modern machine learning applications, such as persistence diagrams (PDs) arising from Topological Data Analysis (TDA), has shown great potential in various domains. However, their practical use in applications is often hindered by two major limitations: the computational complexity required to compute such descriptors exactly, and their sensitivity to even low-level proportions of outliers. In this work, we propose to bypass these two burdens in a data-driven setting by entrusting the estimation of (vectorization of) PDs built on top of point clouds to a neural network architecture that we call RipsNet. Once trained on a given data set, RipsNet can estimate topological descriptors on test data very efficiently with generalization capacity. Furthermore, we prove that RipsNet is robust to input perturbations in terms of the 1-Wasserstein distance, a major improvement over the standard computation of PDs that only enjoys Hausdorff stability, yielding RipsNet to substantially outperform exactly-computed PDs in noisy settings. We showcase the use of RipsNet on both synthetic and real-world data. Our implementation will be made freely and publicly available as part of the open-source library Gudhi.} }
Endnote
%0 Conference Paper %T RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds %A Thibault de Surrel %A Felix Hensel %A Mathieu Carrière %A Théo Lacombe %A Yuichi Ike %A Hiroaki Kurihara %A Marc Glisse %A Frédéric Chazal %B Proceedings of Topological, Algebraic, and Geometric Learning Workshops 2022 %C Proceedings of Machine Learning Research %D 2022 %E Alexander Cloninger %E Timothy Doster %E Tegan Emerson %E Manohar Kaul %E Ira Ktena %E Henry Kvinge %E Nina Miolane %E Bastian Rieck %E Sarah Tymochko %E Guy Wolf %F pmlr-v196-surrel22a %I PMLR %P 96--106 %U https://proceedings.mlr.press/v196/surrel22a.html %V 196 %X The use of topological descriptors in modern machine learning applications, such as persistence diagrams (PDs) arising from Topological Data Analysis (TDA), has shown great potential in various domains. However, their practical use in applications is often hindered by two major limitations: the computational complexity required to compute such descriptors exactly, and their sensitivity to even low-level proportions of outliers. In this work, we propose to bypass these two burdens in a data-driven setting by entrusting the estimation of (vectorization of) PDs built on top of point clouds to a neural network architecture that we call RipsNet. Once trained on a given data set, RipsNet can estimate topological descriptors on test data very efficiently with generalization capacity. Furthermore, we prove that RipsNet is robust to input perturbations in terms of the 1-Wasserstein distance, a major improvement over the standard computation of PDs that only enjoys Hausdorff stability, yielding RipsNet to substantially outperform exactly-computed PDs in noisy settings. We showcase the use of RipsNet on both synthetic and real-world data. Our implementation will be made freely and publicly available as part of the open-source library Gudhi.
APA
de Surrel, T., Hensel, F., Carrière, M., Lacombe, T., Ike, Y., Kurihara, H., Glisse, M. & Chazal, F.. (2022). RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds. Proceedings of Topological, Algebraic, and Geometric Learning Workshops 2022, in Proceedings of Machine Learning Research 196:96-106 Available from https://proceedings.mlr.press/v196/surrel22a.html.

Related Material