WRAP: Wigner Rotation-Equivariant Attention for Physical Signals on The Sphere

Many physical signals are spherical: a radar signature is a function of viewing direction, and an atmospheric state is a field on a rotating planet. Projecting them onto a plane distorts area, distance and orientation and discards the rotational symmetry of the underlying physical system. Previous equivariant attention methods either rely on computationally expensive steerable kernels and tensor products, limiting scalability, or, are restricted to Euclidean space, planar images or camera poses rather than spherical data. We introduce Wigner-RoPE, a spherical rotary positional encoding that generalizes rotary positional embeddings from translations along a line to rotations on the sphere. Wigner-RoPE represents lifted spherical positions as rotations from a reference direction using Wigner D-matrices. Building on Wigner-RoPE, we develop Wigner rotation-equivariant attention for physical signals (WRAP), an expressive, multi-frequency, $SO(3)$- equivariant attention mechanism compatible with efficient fused attention kernels. Stacking WRAP layers yields WRAP-former, a transformer for spherical physical signals. Across radar classification, spherical segmentation, and medium-range weather forecasting, WRAP-former outperforms strong baselines with alternative positional embeddings and symmetry–efficiency tradeoffs. Finally, we compare spherical embedding methods with varying degrees of symmetry breaking, finding that different symmetry choices benefit different physical domains.