# Random Helgason–Fourier Features

Hyperbolic space suits hierarchical embeddings, but earlier hyperbolic Gaussian-process latent variable models had to approximate every kernel entry numerically and optimize auxiliary variables jointly, which became impractical at scale. RHFF-LVM instead approximates hyperbolic kernels with explicit finite-dimensional random features derived from the Helgason–Fourier transform, modeling observations as feature projections of the latents and removing numerical quadrature, variational bounds and inducing points. On a synthetic binary tree, its embedded distances reach a Spearman correlation of 84.3 with true path lengths, versus 74.9 for the Euclidean counterpart RFF-LVM and 82.4 for the strongest hyperbolic baseline, Poincaré Map.

Hyperbolic space suits hierarchical embeddings, but earlier hyperbolic Gaussian-process latent variable models had to approximate every kernel entry numerically and optimize auxiliary variables jointly, which became impractical at scale. RHFF-LVM instead approximates hyperbolic kernels with explicit finite-dimensional random features derived from the Helgason–Fourier transform, modeling observations as feature projections of the latents and removing numerical quadrature, variational bounds and inducing points. On a synthetic binary tree, its embedded distances reach a Spearman correlation of 84.3 with true path lengths, versus 74.9 for the Euclidean counterpart RFF-LVM and 82.4 for the strongest hyperbolic baseline, Poincaré Map.