Hyperbolic PHATE: Visualizing Continuous Hierarchy of Latent Differentiation Structures

This paper proposes a method for embedding diffusion potentials into a hyperbolic space in order to visualize the differentiation structure consisting of diffusion and branching inherent in high-dimensional data. In recent years, the rapid development of single-cell sequencing in the field of biological information processing has made it possible to observe the evolution of gene expression levels in a snapshot-like manner as cells grow from birth to each organ or tissue. Visualization of such high-dimensional (gene pattern dimension) data is expected to provide important insights into the mechanisms of cell differentiation. Therefore, in the visualization of such data, there is a need for a system that emphasizes the "diffusion" structure that gradually shifts with time and the "branching" structure that broadly branches off into individual organs and tissues. Conventionally, the diffusion map and its extension PHATE have been developed as visualization methods specializing in diffusion structures, and hyperbolic embedding has been used as a method specializing in branching structures. However, methods that attempt to explicitly capture diffusion and branching structures simultaneously have not yet received much attention. In this paper, we focus on diffusion mapping (and its extension, PHATE), which specializes in diffusion structures, and hyperbolic embedding, which specializes in branching structures, and propose a visualization method that combines the advantages of both in order to better capture differentiaion structures consisting of diffusion and branching. As a symbolic example, we demonstrate our method using gene-cell expression data in the context of single cell analysis.