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.

Random Fourier features (RFF) represent one of the most popular and wide-spread techniques in machine learning to scale up kernel algorithms. Despite the numerous successful applications of RFFs, unfortunately, quite little is understood theoretically on their optimality and limitations of their performance. To the best of our knowledge, the only existing areas where precise statistical-computational trade-offs have been established are approximation of kernel values, kernel ridge regression, and kernel principal component analysis. Our goal is to spark the investigation of optimality of RFF-based approximations in tasks involving not only function values but derivatives, which naturally lead to optimization problems with kernel derivatives. Particularly, in this paper, we focus on the approximation quality of RFFs for kernel derivatives and prove that the existing finite-sample guarantees can be improved exponentially in terms of the domain where they hold, using recent tools from unbounded empirical process theory. Our result implies that the same approximation guarantee is achievable for kernel derivatives using RFF as for kernel values.