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.

翻译：高斯过程潜在变量模型（GPLVMs）已证明能通过低维表示有效捕捉复杂的高维数据。最新进展表明，使用黎曼流形作为潜在空间为学习更高质量的嵌入提供了更大的灵活性。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。以往方法依赖双曲测地线来插值潜在空间，但这通常导致路径穿越低数据区域，从而产生高度不确定的预测。相反，我们提出通过拉回度量增强双曲度量，以考虑GPLVM非线性映射引入的畸变。通过多项实验，我们证明拉回度量上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。