Poincaré Maps for Analyzing Complex Hierarchies in Single-Cell Data

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 direct exploratory analysis and the use of our embeddings in a wide variety of downstream data analysis tasks, such as visualization, clustering, lineage detection and pseudo-time 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. More specifically, we demonstrate that Poincaré maps allow in a straightforward manner to formulate new hypotheses about biological processes unbeknown to prior methods.Significance statementThe discovery of hierarchies in biological processes is central to developmental biology. We propose Poincaré maps, a new method based on hyperbolic geometry to discover continuous hierarchies from pairwise similarities. We demonstrate the efficacy of our method on multiple single-cell datasets on tasks such as visualization, clustering, lineage identification, and pseudo-time inference.