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

Youssef Wally, Johan Mylius-Kroken, Michael Kampffmeyer, Rezvan Ehsani, Vladan Milosevic, Elisabeth Wetzer

Keywords: Mutual Information, Unsupervised Learning, Hyperbolic Representation, Lorentz Geometry, Imaging Mass Cytometry, Breast Cancer

TL;DR: We propose a geometry-agnostic mutual information framework for evaluating hyperbolic embeddings. Applied to imaging mass cytometry, Lorentzian models align best with biological hierarchies. Code for estimation and visualization is released.

Abstract: 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. Code available at: XXX

Submission Number: 33