Hyperbolic non-metric multidimensional scaling reveals intrinsic geometric structure in high-dimensional data

Modern datasets characterize objects with respect to many variables and assign distances between objects based on a Euclidean metric. However, recent results suggest that for data produced by underlying hierarchical tree-like networks a hyperbolic metric might be more appropriate than a Euclidean one. We develop non-metric multidimensional scaling (MDS) in hyperbolic space to perform hyperbolic embedding of points. Using simulations we show that hyperbolic MDS, combined with Euclidean MDS, can be used to detect intrinsic geometry of data. Applying hyperbolic MDS to human gene expression data, we find that the samples taken from local clusters have Euclidean structure, but samples taken broadly from the whole population show hyperbolic metric, which indicates that the human gene expression space is locally Euclidean but globally hyperbolic. Further we quantify the hyperbolic radii of cells from other diverse biological systems including different mouse organs, finding that mouse brain and embryonic stem cells are also hyperbolic while organs like mouse lung, kidney and placenta are Euclidean. Our method provides a quantitative approach to detecting hidden geometric structures and quantifying cell hierarchies of diverse biological systems.