# Hyperbolic Representation Learning

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.

CRISPR lineage recorders can trace the ancestry of hundreds of thousands of cells, but likelihood-based tree building searches over tree topologies and becomes impractical at that scale. HERACLES places each cell as a point in hyperbolic space, scores the CRISPR editing likelihood of sampled cell quartets, and optimizes those coordinates by gradient descent before a single neighbour-joining pass yields the full tree. On simulated benchmarks its reconstruction error is lower than LAML, Startle-NNI and distance-based methods, and it handles 150,000 cells in about four hours on one GPU.

CRISPR lineage recorders can trace the ancestry of hundreds of thousands of cells, but likelihood-based tree building searches over tree topologies and becomes impractical at that scale. HERACLES places each cell as a point in hyperbolic space, scores the CRISPR editing likelihood of sampled cell quartets, and optimizes those coordinates by gradient descent before a single neighbour-joining pass yields the full tree. On simulated benchmarks its reconstruction error is lower than LAML, Startle-NNI and distance-based methods, and it handles 150,000 cells in about four hours on one GPU.

Studying the implicit bias of Riemannian gradient flow for hyperbolic multiclass classification with fixed prototypes, this work decomposes large-radius distances into a radial term and a Busemann direction term, proving a radial dichotomy in which the sign of a drift coefficient decides whether the radius escapes to the ideal boundary or returns inward, and showing that boundary directions converge to critical points of the Busemann risk, with an asymptotic classifier and decision boundaries on the ideal boundary.

Studying the implicit bias of Riemannian gradient flow for hyperbolic multiclass classification with fixed prototypes, this work decomposes large-radius distances into a radial term and a Busemann direction term, proving a radial dichotomy in which the sign of a drift coefficient decides whether the radius escapes to the ideal boundary or returns inward, and showing that boundary directions converge to critical points of the Busemann risk, with an asymptotic classifier and decision boundaries on the ideal boundary.