用拉回度量改进高维数据的超曲面表示，让预测更准确。
On Probabilistic Pullback Metrics for Latent Hyperbolic Manifolds
- 引入拉回度量修正非线性映射带来的扭曲
- 在超曲面上的最短路径更贴近真实数据分布
- 适合做层次结构建模与不确定性低的生成任务
概率潜在变量模型（LVMs）通过低维表示高效建模高维复杂数据。最近研究发现，为潜在空间赋予黎曼度量可实现符合数据结构的几何距离与最短路径。本文聚焦超曲面嵌入，该结构特别适合建模层次关系。以往依赖超曲面测地线进行潜在空间插值的方法常生成穿越低密度区域的路径，导致预测不确定性高。为此，我们提出在超曲面中引入拉回度量，以补偿LVM非线性映射造成的畸变，并完整推导了高斯过程潜在变量模型（GPLVM）的拉回度量。实验表明，基于拉回度量的测地线不仅符合超曲面几何，还与底层数据分布对齐，显著降低预测不确定性。
原文摘要 · Abstract (English)
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.
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