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.

翻译：非线性降维技术，特别是UMAP，被广泛应用于高维数据的可视化。然而，UMAP的局部欧氏距离假设往往无法捕捉内在流形几何结构，导致拓扑撕裂与结构塌陷。我们发现UMAP对k近邻图的敏感性是其主要原因。为解决这一问题，我们引入Ollivier-Ricci曲率作为几何先验，以强化几何瓶颈处的边并减少冗余连接。由于曲率估计对噪声敏感，我们还通过Jaccard相似度引入拓扑先验以确保邻域一致性。由此得到的方法JORC-UMAP能更好地区分真实流形结构与虚假连接。在合成与真实数据集上的实验表明，以SVM分类精度与三元组保持分数为评价指标，JORC-UMAP相比标准UMAP及其他降维方法能更有效地减少撕裂与塌陷，同时保持计算效率。本研究为UMAP提供了一种几何感知的增强方案，以实现更忠实的数据可视化。