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

• URL: http://arxiv.org/abs/2601.16552v1
• Date: Fri, 23 Jan 2026 08:42:56 GMT
• Title: Understanding and Improving UMAP with Geometric and Topological Priors: The JORC-UMAP Algorithm
• Authors: Xiaobin Li, Run Zhang,
• Abstract summary: dimensionality reduction techniques, particularly UMAP, are widely used for visualizing high-dimensional data.<n>We introduce Ollivier-Ricci curvature as a geometric prior, reinforcing edges at geometric bottlenecks and reducing redundant links.<n>Experiments on synthetic and real-world datasets show that JORC-UMAP reduces tearing and collapse more effectively than standard UMAP and other DR methods.
• Score: 1.7484982792736636
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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