On Probabilistic Pullback Metrics for Latent Hyperbolic Manifolds

Abstract

Probabilistic Latent Variable Models (LVMs) excel at modeling complex, high-dimensional data through lower-dimensional representations. Recent advances show that equipping these latent representations with a Riemannian metric unlocks geometry-aware distances and shortest paths that comply with the underlying data structure. This paper focuses on hyperbolic embeddings, a particularly suitable choice for modeling hierarchical relationships. Previous approaches relying on hyperbolic geodesics for interpolating the latent space often generate paths crossing low-data regions, leading to highly uncertain predictions. Instead, we propose augmenting the hyperbolic manifold with a pullback metric to account for distortions introduced by the LVM's nonlinear mapping and provide a complete development for pullback metrics of Gaussian Process LVMs (GPLVMs). Our experiments demonstrate that geodesics on the pullback metric not only respect the geometry of the hyperbolic latent space but also align with the underlying data distribution, significantly reducing uncertainty in predictions.