# HADES: Fast Singularity Detection with Local Measure Comparison

arXiv: https://arxiv.org/abs/2311.04171 · HTML: https://mlsanity.com/paper/2311.04171/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 53 references that are in the map.
- First posted on arXiv 7 Nov 2023.

## Builds on (most cited first, 2 of 2)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Tangent Space and Dimension Estimation with the Wasserstein Distance](https://mlsanity.com/paper/2110.06357/) (2021)

## Built on by (most cited first, 4 of 4)

- [Token embeddings violate the manifold hypothesis](https://mlsanity.com/paper/2504.01002/) (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)
- [Stratifying Reinforcement Learning with Signal Temporal Logic](https://mlsanity.com/paper/2604.04923/) (2026)

## Abstract (arXiv)

We introduce Hades, an unsupervised algorithm to detect singularities in data. This algorithm employs a kernel goodness-of-fit test, and as a consequence it is much faster and far more scaleable than the existing topology-based alternatives. Using tools from differential geometry and optimal transport theory, we prove that Hades correctly detects singularities with high probability when the data sample lives on a transverse intersection of equidimensional manifolds. In computational experiments, Hades recovers singularities in synthetically generated data, branching points in road network data, intersection rings in molecular conformation space, and anomalies in image data.