On Probabilistic Pullback Metrics on Latent Hyperbolic Manifolds

Gaussian Process Latent Variable Models (GPLVMs) have proven effective in capturing complex, high-dimensional data through lower-dimensional representations. Recent advances show that using Riemannian manifolds as latent spaces provides more flexibility to learn higher quality embeddings. This paper focuses on the hyperbolic manifold, a particularly suitable choice for modeling hierarchical relationships. While previous approaches relied on hyperbolic geodesics for interpolating the latent space, this often results in paths crossing low-data regions, leading to highly uncertain predictions. Instead, we propose augmenting the hyperbolic metric with a pullback metric to account for distortions introduced by the GPLVM's nonlinear mapping. Through various experiments, we 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.