Understanding and Improving UMAP with Geometric and Topological Priors: The JORC-UMAP Algorithm
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911393549123584 |
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| author | Li, Xiaobin Zhang, Run |
| author_facet | Li, Xiaobin Zhang, Run |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_16552 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Understanding and Improving UMAP with Geometric and Topological Priors: The JORC-UMAP Algorithm Li, Xiaobin Zhang, Run Machine Learning Computer Vision and Pattern Recognition Geometric Topology 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. |
| title | Understanding and Improving UMAP with Geometric and Topological Priors: The JORC-UMAP Algorithm |
| topic | Machine Learning Computer Vision and Pattern Recognition Geometric Topology |
| url | https://arxiv.org/abs/2601.16552 |