用几何拓扑先验改进UMAP，减少数据扭曲和结构坍缩。
Understanding and Improving UMAP with Geometric and Topological Priors: The JORC-UMAP Algorithm
- 引入奥利维耶-里奇曲率作为几何先验，强化瓶颈区域边连接
- 结合雅各布相似度拓扑先验，提升邻域一致性，减少冗余连接
- 在保持效率的同时，显著降低数据撕裂与坍缩，适合高维可视化任务
非线性降维方法，尤其是UMAP，广泛用于高维数据可视化。然而，UMAP对局部欧氏距离的假设常无法捕捉流形的内在几何结构，导致拓扑撕裂和结构坍缩。我们发现UMAP对k近邻图的敏感性是关键原因。为此，引入奥利维耶-里奇曲率作为几何先验，增强几何瓶颈处的边连接，减少冗余连接；由于曲率估计易受噪声影响，还结合雅各布相似度构建拓扑先验，确保邻域一致性。提出的JORC-UMAP方法能更好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，相较于标准UMAP及其他降维方法，JORC-UMAP在支持向量机准确率和三元组保持率指标上表现更优，同时维持计算效率。该工作为UMAP提供了几何感知增强，实现更忠实的数据可视化。
原文摘要 · Abstract (English)
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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