Fig. 1: Poincaré maps discover hierarchies and branching processes.
From: Poincaré maps for analyzing complex hierarchies in single-cell data
a Our goal is to recover cell developmental processes, depicted here on the Waddington's epigenetic landscape. b Poincaré disks provide a natural geometry to preserve hierarchical structures and pairwise similarities in two dimensions. Poincaré disks grow as we approach their boundary: all the triangles depicted in the figure are of equal size. c Poincaré maps first estimate geodesic distances, computed from a connected k-nearest-neighbor graph. Second, they compute two-dimensional hyperbolic embeddings that preserve these similarities. d Overview of Poincaré maps embedding procedure. From a given feature matrix, Poincaré maps firsts estimates local similarities based on a user specified local distance metric (Euclidean, cosine, etc.) and Gaussian kernel with a tunable parameter σ. Local similarities then used to compute global proximities on the dataset. By means of Riemanninan optimization of KL divergence, global proximities are aimed to be preserved through global distances in Poincaré disk.
