Hyperbolic Geometry Improves Tree-Structured AI Prototypes

The choice of latent manifold for class prototypes in hierarchical classification models can significantly impact how well tree-structured regularizers are satisfied without distorting data likelihood. This research compares Euclidean (R^d) and hyperbolic (Poincare ball B^d_c) latent geometries, noting that hyperbolic space's exponential volume growth is theoretically better suited for embedding trees with less distortion. Across extensive experiments on the WikiArt dataset, the study reveals a robust finding: Poincare prototypes are substantially better at preserving the topology of the nearest-neighbor graph in latent space than their Euclidean counterparts. This translates to significant improvements in local retrieval tasks, such as sibling and cousin recall. While hyperbolic geometry clearly enhances local topological preservation and local retrieval, the benefits for global tree fidelity comparisons proved unstable across different reference tree definitions, preventing a definitive winner in that aspect.