TOAST: Topological Algorithm for Singularity Tracking

Julius Von Rohrscheidt, Bastian Rieck

Keywords: topology, persistent homology, topological data analysis, tda, stratified spaces, singularities

TL;DR: We develop a multi-scale score that characterises singularities of arbitrary (i.e. non-manifold) data spaces

Abstract: The manifold hypothesis, which assumes that data lie on or close to an unknown manifold of low intrinsic dimensionality, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibit distinct non-manifold structures, which result in singularities that can lead to erroneous conclusions about the data. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address detecting singularities by developing (i) persistent local homology, a new topology-driven framework for quantifying the intrinsic dimension of a data set locally, and (ii) Euclidicity, a topology-based multi-scale measure for assessing the ‘manifoldness’ of individual points. We show that our approach can reliably identify singularities of complex spaces, while also capturing singular structures in real-world data sets.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics

Submission Guidelines: Yes

Please Choose The Closest Area That Your Submission Falls Into: Theory (eg, control theory, learning theory, algorithmic game theory)

Community Implementations: [ 2 code implementations](https://www.catalyzex.com/paper/arxiv:2210.00069/code)