# Gaussian Process Latent Variable Model

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.

This work applies amortized structured stochastic variational inference to Gaussian process latent variable models so that the variational posterior over the latent space depends conditionally on the values of the inducing points, moving beyond the mean-field approximation between inducing points and latent variables, and reports that this more flexible variational posterior improves several metrics relating to the reconstruction of points on the data manifold.

This work applies amortized structured stochastic variational inference to Gaussian process latent variable models so that the variational posterior over the latent space depends conditionally on the values of the inducing points, moving beyond the mean-field approximation between inducing points and latent variables, and reports that this more flexible variational posterior improves several metrics relating to the reconstruction of points on the data manifold.