On Probabilistic Pullback Metrics for Latent Hyperbolic Manifolds

TL;DR

The paper tackles interpolation in hyperbolic latent spaces used to model hierarchical data, showing that standard hyperbolic geodesics can pass through low-data regions and increase uncertainty. It introduces a hyperbolic pullback metric induced by a stochastic immersion $f:\,\mathbb{H}^{D_x}_{\mathcal{L}} \to \mathbb{R}^{D_y}$, with the latent-space metric $\bm{G}^{\text{P}}_{\bm{x}}=\tilde{\bm{J}}^\top \bm{G}_{\bm{y}} \tilde{\bm{J}}$ and an expectation $\mathbb{E}[\bm{G}^{\text{P}}_{\bm{x}}] = \mathbb{E}[\bm{J}]^\top \mathbb{E}[\bm{J}] + D_y \bm{\Sigma}_J$, enabling geometry-aware distances. It then specializes to the gphlvm by deriving the Jacobian distribution under GP conditioning, yielding $\mathbb{E}[\bm{G}^{\text{P},\mathcal{L}}_{\bm{x}^*}] = \bm{P}_{\bm{x}^*}(\boldsymbol{\mu}_J^\top \boldsymbol{\mu}_J + D_y \boldsymbol{\Sigma}_J) \bm{P}_{\bm{x}^*}^\top$, where the projection $\bm{P}_{\bm{x}^*}$ enforces hyperbolic tangent-space constraints. Hyperbolic pullback geodesics are computed by minimizing the curve energy with the pullback metric, including a spline regularization, using a Riemannian optimizer. The work also provides analytic derivatives for 2D and 3D hyperbolic SE kernels to enable efficient gradient-based optimization, and demonstrates through four experiments that pullback geodesics follow data support and reduce uncertainty compared with baseline geodesics, highlighting practical gains in manifold-aware latent distances and generative 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.

Figures (10)

• Figure 1: Hyperbolic pullback metric on the tangent space of the Lorentz model. The pullback geodesic () follows the data () manifold, in contrast to the hyperbolic geodesic ().
• Figure 2: Left and middle: Euclidean and hyperbolic pullback metrics on an embedded $\mathsf{C}$-shape trajectory () with base manifold () and pullback () geodesics. Right: Curve energy along the geodesics on the Euclidean (top) and hyperbolic (bottom) cases. These include the base manifold geodesic (), the base manifold geodesic with energy \ref{['eq:curve_energy']} evaluated using the pullback metric (), and the pullback geodesic ().
• Figure 3: Left: Embeddings of a subset of the MNIST dataset with digits $\mathsf{0}$ (), $\mathsf{1}$ (), $\mathsf{2}$ (), $\mathsf{3}$ (), $\mathsf{6}$ (), and $\mathsf{9}$ (). The background color represents the pullback metric volume. The base manifold () and pullback () geodesics interpolate between a $\mathsf{3}$ and a $\mathsf{6}$. Right: Ten samples along the decoded geodesics in image space.
• Figure 4: Left: Embeddings of multi-cellular robots from coarse (darker tone) to fine (lighter tone). The embeddings form two clusters originating from an all-vertically-actuated robot () and an all-horizontally-actuated robot (). Right: Ten samples along the decoded base manifold () and pullback () geodesics.
• Figure 5: Top left: Embeddings of hand grasps colored according to their corresponding grasp class. The background color represents the pullback metric volume. The base manifold () and pullback () geodesics correspond to a transition from a ring () to a spherical grasp (). Top right: Time-series plots of $2$ dimensions of the joint space showing the mean of the decoded geodesics with their uncertainty as a gray envelope. A training trajectory to the spherical grasp () and a reversed training trajectory from the ring grasp () are included for reference. Bottom: Generated hand trajectories from the decoded geodesics.