Topological Singularity Detection at Multiple Scales

Julius Von Rohrscheidt, Bastian Rieck
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:35175-35197, 2023.

Abstract

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the ’manifoldness’ of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-von-rohrscheidt23a, title = {Topological Singularity Detection at Multiple Scales}, author = {Von Rohrscheidt, Julius and Rieck, Bastian}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {35175--35197}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/von-rohrscheidt23a/von-rohrscheidt23a.pdf}, url = {https://proceedings.mlr.press/v202/von-rohrscheidt23a.html}, abstract = {The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the ’manifoldness’ of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.} }
Endnote
%0 Conference Paper %T Topological Singularity Detection at Multiple Scales %A Julius Von Rohrscheidt %A Bastian Rieck %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-von-rohrscheidt23a %I PMLR %P 35175--35197 %U https://proceedings.mlr.press/v202/von-rohrscheidt23a.html %V 202 %X The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the ’manifoldness’ of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.
APA
Von Rohrscheidt, J. & Rieck, B.. (2023). Topological Singularity Detection at Multiple Scales. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:35175-35197 Available from https://proceedings.mlr.press/v202/von-rohrscheidt23a.html.

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