  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.
