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

TL;DR

This paper introduces JORC-UMAP, an enhanced dimensionality reduction method that incorporates geometric and topological priors to better preserve manifold structures and improve data visualization quality.

Contribution

The paper proposes JORC-UMAP, a novel UMAP variant that integrates Ollivier-Ricci curvature and Jaccard similarity to address UMAP's limitations in capturing intrinsic manifold geometry.

Findings

JORC-UMAP reduces topological tearing and collapse more effectively than standard UMAP.

It maintains computational efficiency while improving manifold structure preservation.

Experiments show enhanced visualization quality on synthetic and real datasets.

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.