  We propose a novel quaternion product unit (QPU) to represent data on 3D
rotation groups. The QPU leverages quaternion algebra and the law of 3D
rotation group, representing 3D rotation data as quaternions and merging them
via a weighted chain of Hamilton products. We prove that the representations
derived by the proposed QPU can be disentangled into "rotation-invariant"
features and "rotation-equivariant" features, respectively, which supports the
rationality and the efficiency of the QPU in theory. We design quaternion
neural networks based on our QPUs and make our models compatible with existing
deep learning models. Experiments on both synthetic and real-world data show
that the proposed QPU is beneficial for the learning tasks requiring rotation
robustness.
