Poincaré maps for analyzing complex hierarchies in single-cell data

Anna Klimovskaia, David Lopez-Paz, Léon Bottou, Maximilian Nickel

Abstract:

The need to understand cell developmental processes spawned a plethora of computational
methods for discovering hierarchies from scRNAseq data. However, existing techniques are
based on Euclidean geometry, a suboptimal choice for modeling complex cell trajectories with
multiple branches. To overcome this fundamental representation issue we propose Poincaré
maps, a method that harness the power of hyperbolic geometry into the realm of single-cell
data analysis. Often understood as a continuous extension of trees, hyperbolic geometry
enables the embedding of complex hierarchical data in only two dimensions while preserving
the pairwise distances between points in the hierarchy. This enables the use of our
embeddings in a wide variety of downstream data analysis tasks, such as visualization,
clustering, lineage detection and pseudotime inference. When compared to existing methods
— unable to address all these important tasks using a single embedding — Poincaré maps
produce state-of-the-art two-dimensional representations of cell trajectories on multiple
scRNAseq datasets.