Riemannian Isometric Kernels: Preserving Intrinsic Geometry of Hyperbolic Space

Hyperbolic spaces, characterized by constant negative curvature, provide a natural geometric alternative for representing hierarchical data, while kernel methods further enhance the representation power by mapping hyperbolic embeddings into reproducing kernel Hilbert spaces (RKHSs). However, existing hyperbolic kernels enforce the isometry constraint only at the metric-space level, creating a substantial mismatch between the kernel-induced similarity used in practice and the manifold's theoretical geodesic distance. To address this limitation, we introduce Riemannian isometric kernels, whose canonical feature maps induce a metric that coincides exactly with the Riemannian metric of the input manifold, thereby intrinsically preserving its geometric structure. Starting from the definition of Riemannian isometry, we construct a family of Riemannian isometric kernels, including logarithmic kernel (RILog), binomial kernel (RIBin), exponential kernel (RIExp), and integral kernel (RIInt) variants, whose kernel-induced distances used in practice are monotonically increasing functions of the geodesic distance, ensuring consistency between the theoretical formulation and empirical evaluation. We evaluate the proposed kernels through geometric stress analysis and four diverse downstream tasks: zero-shot learning, few-shot learning, graph node classification, and semantic textual similarity. Across all benchmarks, the proposed kernels match or outperform state-of-the-art adaptive hyperbolic kernels, demonstrating their effectiveness and broad applicability.