Multiscale Hyperbolic Embedding Reveals Hierarchical Structure in Complex Biological Systems

The rapid expansion of biological and computational datasets demands scalable methods that support both visualization and quantitative interpretation. Hyperbolic embeddings are well suited to represent hierarchical structure, but existing approaches are limited by fixed curvature assumptions or poor scalability to large datasets. We introduce MuH-MDS, a multiscale hyperbolic multidimensional scaling algorithm that employs an adi-abatic optimization strategy: local positions are iteratively refined while cluster centroids are temporarily fixed. This strategy accelerates computation by 103 and enables scaling to datasets with over 80,000 samples. Applied to diverse benchmarks, including C. elegans embryogenesis scRNA-seq data, MuH-MDS uncovers intrinsic hierarchical organization and improves both pseudotime inference and lineage reconstruction relative to UMAP and other standard methods. In contrast to UMAP and t-SNE, which prioritize local neighborhoods at the expense of global coherence and metric fidelity, MuH-MDS pre-serves both local detail and global hierarchy, providing a metrically faithful framework for multiscale analysis of complex biological systems.