Global and Relative Topological Features from Homological Invariants of Subsampled Datasets

Jens Agerberg, Wojciech Chacholski, Ryan Ramanujam
Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), PMLR 221:302-312, 2023.

Abstract

Homology-based invariants can be used to characterize the geometry of datasets and thereby gain some understanding of the processes generating those datasets. In this work we investigate how the geometry of a dataset changes when it is subsampled in various ways. In our framework the dataset serves as a reference object; we then consider different points in the ambient space and endow them with a geometry defined in relation to the reference object, for instance by subsampling the dataset proportionally to the distance between its elements and the point under consideration. We illustrate how this process can be used to extract rich geometrical information, allowing for example to classify points coming from different data distributions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v221-agerberg23a, title = {Global and Relative Topological Features from Homological Invariants of Subsampled Datasets}, author = {Agerberg, Jens and Chacholski, Wojciech and Ramanujam, Ryan}, booktitle = {Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML)}, pages = {302--312}, year = {2023}, editor = {Doster, Timothy and Emerson, Tegan and Kvinge, Henry and Miolane, Nina and Papillon, Mathilde and Rieck, Bastian and Sanborn, Sophia}, volume = {221}, series = {Proceedings of Machine Learning Research}, month = {28 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v221/agerberg23a/agerberg23a.pdf}, url = {https://proceedings.mlr.press/v221/agerberg23a.html}, abstract = {Homology-based invariants can be used to characterize the geometry of datasets and thereby gain some understanding of the processes generating those datasets. In this work we investigate how the geometry of a dataset changes when it is subsampled in various ways. In our framework the dataset serves as a reference object; we then consider different points in the ambient space and endow them with a geometry defined in relation to the reference object, for instance by subsampling the dataset proportionally to the distance between its elements and the point under consideration. We illustrate how this process can be used to extract rich geometrical information, allowing for example to classify points coming from different data distributions.} }
Endnote
%0 Conference Paper %T Global and Relative Topological Features from Homological Invariants of Subsampled Datasets %A Jens Agerberg %A Wojciech Chacholski %A Ryan Ramanujam %B Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML) %C Proceedings of Machine Learning Research %D 2023 %E Timothy Doster %E Tegan Emerson %E Henry Kvinge %E Nina Miolane %E Mathilde Papillon %E Bastian Rieck %E Sophia Sanborn %F pmlr-v221-agerberg23a %I PMLR %P 302--312 %U https://proceedings.mlr.press/v221/agerberg23a.html %V 221 %X Homology-based invariants can be used to characterize the geometry of datasets and thereby gain some understanding of the processes generating those datasets. In this work we investigate how the geometry of a dataset changes when it is subsampled in various ways. In our framework the dataset serves as a reference object; we then consider different points in the ambient space and endow them with a geometry defined in relation to the reference object, for instance by subsampling the dataset proportionally to the distance between its elements and the point under consideration. We illustrate how this process can be used to extract rich geometrical information, allowing for example to classify points coming from different data distributions.
APA
Agerberg, J., Chacholski, W. & Ramanujam, R.. (2023). Global and Relative Topological Features from Homological Invariants of Subsampled Datasets. Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), in Proceedings of Machine Learning Research 221:302-312 Available from https://proceedings.mlr.press/v221/agerberg23a.html.

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