RAP: Radial-Angular Random Projection for Hyperbolic Representations

Random projection is a basic compression primitive for Euclidean representation learning, with the Johnson–Lindenstrauss lemma giving a target dimension $k=O({\epsilon }^{-2}\log n)$ independent of the ambient dimension. As hyperbolic representations gain increasing attention in modern Transformers, vision-language models, and large language models, the same need for oblivious dimensionality reduction arises, but no analogous primitive yet exists for hyperbolic representations. We study this gap. Two natural attempts fail in dual ways: a tangent-space scheme gives a clean local guarantee but is chart-bound and pays a curvature penalty, while a fully hyperbolic projection in the hyperboloid model is geometrically native but mixes radial and angular error before feeding them into the conditioning of $\mathrm{arcosh}$. We propose radial-angular random projection (RAP), which preserves the radial coordinate exactly and applies a Euclidean random projection only to the angular direction, lifting the result back to the target hyperboloid. RAP is hyperbolic-to-hyperbolic by construction, costs one Gaussian matrix-vector product, and reduces hyperbolic compression to a normalized angular JL question on ${\mathbb{S}}^{d-1}$. We prove a finite-set RAP guarantee with explicit $\sinh (\sqrt{c}{r}_x)\sinh (\sqrt{c}{r}_y)$ conditioning, complemented by a local tangent-space JL theorem and an arcosh-conditioning lemma for the fully hyperbolic comparator. Across an origin-centered broad-radius point cloud, a balanced tree, a clustered Gaussian, and the WordNet noun hierarchy, RAP attains the lowest mean relative distortion in every $(n,k)$ cell, reducing distortion over the strongest hyperbolic baseline by roughly an order of magnitude on broad-radius data. RAP also scales efficiently to million-point datasets and delivers improved retrieval performance on vision–language representations. Code is available in an anonymous repository: https://anonymous.4open.science/r/RAP-0C10/README.md.