Topological Singularity Detection at Multiple Scales
This addresses a fundamental issue in machine learning for researchers and practitioners by providing a method to detect data singularities, which is crucial for accurate interpolation and inference, though it appears incremental as it builds on existing topological concepts.
The paper tackles the problem of detecting non-manifold structures (singularities) in data, which can cause errors in machine learning tasks, by developing a topological framework that quantifies local intrinsic dimension and Euclidicity scores across multiple scales, enabling identification of singularities in complex spaces and image data.
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.