Helgason-Fourier Random Features for Scalable Hyperbolic Latent Variables
Koshi Watanabe, Daichi Mochihashi, Hideitsu Hino, Takahiro Ogawa, Miki Haseyama
原文摘要
Hyperbolic geometry is well suited to embedding hierarchical data, capturing tree-like structure faithfully even in the low dimensions where Euclidean dimensionality reduction is severely constrained. We introduce Random Helgason--Fourier Features (RHFF), an explicit finite-dimensional feature map for kernels on hyperbolic space. We characterize the spectral density from which RHFF frequencies are drawn, allowing kernels such as the Matérn family to be specified directly in the frequency domain. We also prove a uniform approximation error bound for RHFF. Building on this representation, we develop a hyperbolic latent variable model called RHFF-LVM, which requires no numerical quadrature, variational bounds or auxiliary variables unlike previous hyperbolic GP-LVMs. Our RHFF-LVM recovers the metric of a synthetic tree more faithfully than all baselines and attains the best mean rank, driven by large gains on tree-structured atlases at $Q\!=\!2$. Furthermore, it fits $5\times10^{5}$ points in $505$ seconds, making it the first non-parametric hyperbolic LVM to reach this scale.
cs.LG