Multiscale Hyperbolic Embedding for Cell Hierarchies in Large-Scale Bioinformatics Data

The increasing size of datasets poses challenges for their visualization and interpretation, highlighting the need for scalable and effective analysis methods. Hyperbolic embedding have shown strong potential in capturing complex hierarchical structures across diverse systems. However, existing hyperbolic embedding methods typically operate with fixed curvature and have difficulties scaling to large datasets. To address these limitations, we propose MuH-MDS, a novel multiscale algorithm for hyperbolic multidimensional scaling that uses "adiabatic" approximation from physics to optimize local positions while keeping cluster centroid fixed. MuH-MDS improves computing time by 10³ compared to previous methods and is able to handle large datasets comprising over 80, 000 samples. We validate the method on a number of datasets, including a large-scale C. elegans embryogenesis scRNA-seq dataset with over 80,000 samples. Here, MuH-MDS uncovers intrinsic hierarchical structure, and achieves improved pseudotime inference and lineage analysis compared to UMAP and other methods. Unlike UMAP and t-SNE, which emphasize local structure at the expense of global coherence and metric accuracy, MuH-MDS preserves global hierarchy in a metrically faithful manner, maintaining key relationships across scales.