CAT: Curvature-Adaptive Transformers for Geometry-Aware Learning
This addresses the problem of handling mixed geometric properties in data for researchers and practitioners in relational reasoning, though it is an incremental advance over fixed-geometry approaches.
The paper tackles the limitation of transformers assuming Euclidean geometry by introducing the Curvature-Adaptive Transformer (CAT), which dynamically routes tokens across hyperbolic, spherical, and Euclidean attention branches, achieving approximately 10% improvements in MRR and Hits@10 on knowledge graph completion benchmarks with minimal overhead.
Transformers achieve strong performance across diverse domains but implicitly assume Euclidean geometry in their attention mechanisms, limiting their effectiveness on data with non-Euclidean structure. While recent extensions to hyperbolic and spherical spaces show promise for hierarchical and cyclical patterns, respectively, they require committing to a single geometry a priori, reducing flexibility when data exhibits mixed geometric properties. We introduce the Curvature-Adaptive Transformer (CAT), a novel architecture that dynamically learns per-token routing across three geometric attention branches through a lightweight, differentiable gating mechanism. Unlike fixed-geometry approaches, CAT enables adaptive geometric specialization, routing tokens to the appropriate curvature based on their local relational structure. The routing network provides interpretable curvature preferences while each branch employs geometry-specific operations optimized for its respective manifold. On knowledge graph completion benchmarks (FB15k-237, WN18RR), CAT achieves approximately 10% improvements in MRR and Hits@10 over fixed-geometry baselines with minimal overhead (5% parameter increase, comparable inference time). These results demonstrate that learned geometric adaptation outperforms any single fixed geometry for complex relational reasoning, establishing CAT as a scalable and interpretable foundation for mixture-of-geometry architectures across language, vision, and multimodal domains.