On Probabilistic Pullback Metrics for Latent Hyperbolic Manifolds

TL;DR

This paper introduces a novel pullback metric for hyperbolic latent spaces in probabilistic models, improving the alignment of geodesics with data structure and reducing uncertainty in predictions.

Contribution

It develops a comprehensive framework for pullback metrics in Gaussian Process LVMs on hyperbolic manifolds, enhancing geometry-aware interpolation.

Findings

Pullback geodesics better respect hyperbolic geometry.

Reduced uncertainty in data predictions.

Improved data-aligned latent space interpolations.

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.