Rethinking the Fully Hyperbolic Vision Transformer in Polar Coordinates

Ahmad Bdeir, Niels Landwehr

Hyperbolic space can embed intrinsic hierarchies in data with low distortion due to the exponential growth of volume with distance from the origin. However, current Lorentz transformer blocks are formulated in ambient coordinates, where numerical errors increase at large radii due to instability in the Lorentzian inner product, leading many models to limit the radius to avoid this issue. This prevents us from utilizing the regions of hyperbolic space that motivate the geometry. To address this, we revisit the components of the transformer block in polar coordinates, where hyperbolic operations such as distance calculation and attention centroids can be computed without the numerical cancellation in their ambient-coordinate formulations. Specifically, we propose a polar fully connected layer that separately maps an embedding's direction and radius, allowing the radius of a feature to be learned rather than determined by the norm of a linear map. We additionally introduce horospherical shifts as relative positional encodings whose query-key distances grow logarithmically with the token gap. Finally, we reformulate the residual connection as average radius Lorentz boosts. Combining…