# Topological Singularity Detection at Multiple Scales

arXiv: https://arxiv.org/abs/2210.00069 · HTML: https://mlsanity.com/paper/2210.00069/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 30 Sep 2022.

## Builds on (most cited first, 14 of 14)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration](https://mlsanity.com/paper/2004.02551/) (2020)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [DTM-based Filtrations](https://mlsanity.com/paper/1811.04757/) (2018)
- [Topology of Word Embeddings: Singularities Reflect Polysemy](https://mlsanity.com/paper/2011.09413/) (2020)
- [On the Need for Topology-Aware Generative Models for Manifold-Based Defenses](https://mlsanity.com/paper/1909.03334/) (2019)
- [FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles](https://mlsanity.com/paper/2206.06513/) (2022)
- [Diffusion Curvature for Estimating Local Curvature in High Dimensional Data](https://mlsanity.com/paper/2206.03977/) (2022)
- [Dimension Detection with Local Homology](https://mlsanity.com/paper/1405.3534/) (2014)
- [Persistent Intersection Homology for the Analysis of Discrete Data](https://mlsanity.com/paper/1907.13485/) (2019)

## Built on by (most cited first, 11 of 11)

- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Token embeddings violate the manifold hypothesis](https://mlsanity.com/paper/2504.01002/) (2025)
- [The structure of the token space for large language models](https://mlsanity.com/paper/2410.08993/) (2024)
- [Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms](https://mlsanity.com/paper/2410.02622/) (2024)
- [Mapping the Multiverse of Latent Representations](https://mlsanity.com/paper/2402.01514/) (2024)
- [Exploring the Stratified Space Structure of an RL Game with the Volume Growth Transform](https://mlsanity.com/paper/2507.22010/) (2025)
- [Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures](https://mlsanity.com/paper/2606.19894/) (2026)
- [Recovering Weighted Tangent Geometry from a Single-Scale Score Field](https://mlsanity.com/paper/2608.22334/) (2026)
- [Topological Inductive Bias fosters Multiple Instance Learning in Data-Scarce Scenarios](https://mlsanity.com/paper/2307.14025/) (2023)
- [Concept Boundary Vectors](https://mlsanity.com/paper/2412.15698/) (2024)

## Abstract (arXiv)

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.