H-KAN: Hyperbolic Kuramoto Attention Networks for Hierarchical Representation Learning

Deep neural networks operating in Euclidean space face a fundamental geometric limitation when representing hierarchical data: while tree-structured information grows exponentially with depth, Euclidean volume grows only polynomially, forcing either intractable dimensionality or catastrophic metric distortion. Concurrently, distributed representations lack robust mechanisms for temporal binding—dynamically grouping features that belong together while segregating unrelated information. We introduce H-KAN (Hyperbolic Kuramoto Attention Networks), an architecture that addresses both challenges by coupling hyperbolic geometry with oscillatory synchronization dynamics. H-KAN embeds neural cellular automata in the Poincaré disk, where exponential volume growth naturally accommodates hierarchical structures. Temporal binding is achieved through Hyperbolic Artificial Kuramoto Oscillatory Neurons (H-AKOrN), extending recent work on oscillatory networks to negatively curved manifolds with geodesic-modulated coupling. A geodesic attention mechanism integrates features based on both semantic similarity and hyperbolic distance, while topological regularization via persistent homology encourages structurally coherent representations. We evaluate H-KAN on hierarchical embedding (WordNet), topological preservation under noise, and temporal binding (Cluttered MNIST). H-KAN achieves +24% Mean Average Precision over Euclidean baselines on WordNet hierarchy reconstruction, +26% Betti number recovery accuracy under high noise (σ=2.0\sigma=2.0 σ=2.0), and +8% Adjusted Rand Index on object segmentation compared to prior oscillatory methods. Systematic ablations confirm that hyperbolic geometry, oscillatory dynamics, and topological regularization provide complementary benefits. CKA-based divergence analysis demonstrates that geometric constraints induce qualitatively distinct learning trajectories rather than merely quantitative improvements. H-KAN provides a principled framework for learning representations of hierarchical and temporally structured data, with documented trade-offs in computational overhead (67% relative to non-oscillatory hyperbolic methods) that inform practical deployment decisions.