Flatland and Beyond: Mutual Information Across Geometries

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

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 shown compelling advantages over conventional Euclidean representation learning in modeling 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 composition 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 neighbor estimators to rigorously quantify the clustering performance in each geometry using mutual information and conditional mutual information. Our results reveal that hyperbolic embeddings retain significantly more biologically relevant information than their Euclidean counterparts. 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 hyperbolic 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: \url{https://github.com/****/****}

Track: Full paper (8 pages excluding references, same as main conference requirements)

Submission Number: 7