Hyperbolic dimensionality reduction drops variational bounds: RHFF-LVM's closed-form random features reach 84.3 tree-distance correlation on a synthetic tree
Synopsis
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.
Interpretation
A hyperbolic latent variable model can now be fit with a fixed set of explicit random features, without numerically integrating each kernel entry or jointly optimizing inducing points. Earlier hyperbolic Gaussian-process latent variable models relied on a variational lower bound whose tightness depends on auxiliary variables optimized with the latents, and the existing implementation fixed its inducing inputs rather than optimizing them. On a synthetic binary tree, RHFF-LVM (heat) reaches a Spearman correlation of 84.3 between embedded distances and true path lengths, against 74.9 for the Euclidean counterpart RFF-LVM and 82.4 for the strongest external hyperbolic baseline, Poincaré Map.
The features have closed forms in terms of Busemann functions, and frequencies are drawn from the Helgason–Fourier spectral density, so hyperbolic kernels such as the heat and Matérn families can be specified directly in the frequency domain. Euclidean random features rely on Bochner's theorem and translation invariance, neither of which holds in hyperbolic space, so the construction moves to the spherical transform and ideal directions on the boundary. Across eight real datasets, RHFF-LVM (heat) attains the best mean rank of 2.55 on the global-structure metric, ahead of PCA at 3.35, and scores 78.8 on C. elegans versus 71.0 for PCA.
Latents are reparameterized as unconstrained Euclidean coordinates and updated by natural gradient descent under the pullback metric, so no tangent-space projections or retractions are needed. Riemannian updates directly on the Lorentz model accumulate floating-point error that grows with distance from the origin and destabilizes training. In a scaling experiment that increases only samples per node of the synthetic tree, RHFF-LVM scales near-linearly and fits half a million points in seconds, while BHGP-LVM cannot scale further because of its variational inference and Poincaré Map is limited by its dense affinity matrix.
Perspective
The demonstrated setting is a synthetic binary tree, seven single-cell atlases and MNIST as a non-hierarchical control, at latent dimensions 2, 5 and 10, scored by global distance correlation and nearest-neighbor recall; the scaling experiment varies only samples per node, up to half a million points.
Approximation error grows exponentially with latent dimension and domain diameter, restricting the method to low-dimensional, bounded regions; peak memory is dominated by the feature matrix, and mini-batch training is left to future work; against PCA the global metric splits three wins to five losses across eight datasets, and the across-dataset test is not significant.