UC San Diego

Many biological systems have intrinsic hierarchical structure, which can be best described by hyperbolic geometry. Hyperbolic geometry was well developed in mathematics, and has recently been introduced in machine learning for data representation and visualization. However, the relevance of hyperbolic geometry to biological data was rarely studied. In this thesis, I systematically detect the existence of hyperbolic geometry in various biological systems, and show the advantages of hyperbolic representation of biological data. The geometry detection of these data are performed using either the fine-but-slow Betti curve method or the rough-but- fast multi-dimensional scaling method. It turns out that many biological systems, including olfactory data from different species and gene expression profiles from different sources, all have hyperbolic structure. The visualization of the data can be achieved by using either hyperbolic multi-dimensional scaling or hyperbolic t-SNE algorithm. The former method correctly reveals the global organization of data points and determines the phenotype related axes in the space, e.g. the pleasantness axis in olfactory space. The latter method aims to perform local clustering as well as preserving inter-cluster structure, giving an accurate representation of different cell types in an informative and organized way. In addition to the improved data visualization, hyperbolic representation of data can inspire new discoveries that cannot be obtained by other Euclidean-based methods, for example, different cell types are characterized by distinct spatial patterns of points in the hyperbolic disk, regardless of brain regions.