Extracting Knowledge from Spatial Biology: Evaluating Cell Type Hierarchies in Breast Cancer Imaging Data

Hyperbolic representation learning has shown compelling advantages over conventional Eu- clidean representation learning in modelling hierarchical relationships in data. In this work, we evaluate its potential to capture biological relations between cell types in highly multiplexed imaging data, where capturing subtle, hierarchical relationships between cell types is crucial to understand tissue composi- tion and functionality. Using a recent and thoroughly validated 42-marker Imaging Mass Cytometry (IMC) dataset of breast cancer tissue, we embed cells into both Euclidean and Lorentzian latent spaces via a fully hyperbolic variational autoencoder. We then introduce an information-theoretic framework based on k-nearest neighbour estimators to rigorously quantify the clustering performance in each geom- etry using mutual information and conditional mutual information. Our results reveal that hyperbolic embeddings retain significantly more biologically relevant information than their Euclidean counter- parts. We further provide open-source tools to extend Kraskov-Stögbauer-Grassberger based mutual information estimation to Lorentzian geodesic spaces, and to enable UMAP visualizations with hyper- bolic distance metrics. This work contributes a principled evaluation method for geometry-aware learning and supports the growing evidence of hyperbolic geometry’s benefits in spatial biology. Code is
available at: https://github.com/youssefwally/FlatlandandBeyond