Hyperbolic Representation Learning for Spatial Biology: Capturing Cell Type Hierarchies in Breast Cancer

Hyperbolic representation learning has recently emerged as a powerful framework for modeling hierarchical structures in data, often outperforming Euclidean embeddings. We investigate its utility for analyzing high-dimensional biological data from Imaging Mass Cytometry (IMC) of breast cancer tissues. We embed cells into Euclidean and Lorentzian latent spaces via a fully hyperbolic variational autoencoder (VAE) and propose an information-theoretic framework based on k-nearest neighbor estimators to quantify clustering quality using mutual information (MI) and conditional mutual information (CMI). Results show that Lorentzian embeddings preserve substantially more biologically relevant structure compared to Euclidean ones. We further provide open-source tools for Lorentzian MI estimation and hyperbolic UMAP visualization, enabling geometry-aware representation learning for spatial biology.