  Rotation equivariance is a desirable property in many practical applications
such as motion forecasting and 3D perception, where it can offer benefits like
sample efficiency, better generalization, and robustness to input
perturbations. Vector Neurons (VN) is a recently developed framework offering a
simple yet effective approach for deriving rotation-equivariant analogs of
standard machine learning operations by extending one-dimensional scalar
neurons to three-dimensional "vector neurons." We introduce a novel
"VN-Transformer" architecture to address several shortcomings of the current VN
models. Our contributions are: $(i)$ we derive a rotation-equivariant attention
mechanism which eliminates the need for the heavy feature preprocessing
required by the original Vector Neurons models; $(ii)$ we extend the VN
framework to support non-spatial attributes, expanding the applicability of
these models to real-world datasets; $(iii)$ we derive a rotation-equivariant
mechanism for multi-scale reduction of point-cloud resolution, greatly speeding
up inference and training; $(iv)$ we show that small tradeoffs in equivariance
($\epsilon$-approximate equivariance) can be used to obtain large improvements
in numerical stability and training robustness on accelerated hardware, and we
bound the propagation of equivariance violations in our models. Finally, we
apply our VN-Transformer to 3D shape classification and motion forecasting with
compelling results.
