Probabilistic Latent Variable Models (LVMs) 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. 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 metric 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.

翻译：概率潜在变量模型（LVMs）已被证明能够通过低维表示有效捕捉复杂的高维数据。最新研究表明，使用黎曼流形作为潜在空间能够为学习更高质量的嵌入提供更大的灵活性。本文聚焦于双曲流形，这是一种特别适用于建模层次关系的选择。以往依赖双曲测地线插值潜在空间的方法，其生成的路径常穿越低数据区域，导致预测结果具有高度不确定性。为此，我们提出通过引入拉回度量来增强双曲度量，以考虑LVM非线性映射引入的畸变，并为高斯过程潜在变量模型（GPLVMs）的拉回度量提供了完整的理论推导。我们的实验表明，基于拉回度量的测地线不仅尊重双曲潜在空间的几何结构，还与底层数据分布保持一致，从而显著降低了预测的不确定性。