# Understanding and Improving UMAP with Geometric and Topological Priors: The JORC-UMAP Algorithm

arXiv: https://arxiv.org/abs/2601.16552 · HTML: https://mlsanity.com/paper/2601.16552/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 32 references that are in the map.
- First posted on arXiv 23 Jan 2026.

## Builds on (most cited first, 8 of 8)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [Revisiting Over-smoothing and Over-squashing Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2211.15779/) (2022)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [TriMap: Large-scale Dimensionality Reduction Using Triplets](https://mlsanity.com/paper/1910.00204/) (2019)
- [Recovering Manifold Structure Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2410.01149/) (2024)

## Abstract (arXiv)

Nonlinear dimensionality reduction techniques, particularly UMAP, are widely used for visualizing high-dimensional data. However, UMAP's local Euclidean distance assumption often fails to capture intrinsic manifold geometry, leading to topological tearing and structural collapse. We identify UMAP's sensitivity to the k-nearest neighbor graph as a key cause. To address this, we introduce Ollivier-Ricci curvature as a geometric prior, reinforcing edges at geometric bottlenecks and reducing redundant links. Since curvature estimation is noise-sensitive, we also incorporate a topological prior using Jaccard similarity to ensure neighborhood consistency. The resulting method, JORC-UMAP, better distinguishes true manifold structure from spurious connections. Experiments on synthetic and real-world datasets show that JORC-UMAP reduces tearing and collapse more effectively than standard UMAP and other DR methods, as measured by SVM accuracy and triplet preservation scores, while maintaining computational efficiency. This work offers a geometry-aware enhancement to UMAP for more faithful data visualization.