Before revision:  In addition to this classification, we use Mackey theory to show that convolutions with equivariant kernels are the most general class of equivariant maps between such fields, thus establishing G-CNNs as a universal class of equivariant networks. The theory also explains how the space of equivariant kernelscan be parameterized for learning, thereby simplifying the development of G-CNNs for new spaces and symmetries. Finally, the theory introduces a rich geometric semantics to learned feature spaces, thus improving interpretability of deep networks, and establishing a connection to central ideas in mathematics and physics .
After revision:  In addition to this classification, we use Mackey theory to show that convolutions with equivariant kernels are the most general kind of equivariant linear map between feature spaces (fields) of given types? Following Mackey, we show that such maps correspond one-to-one with convolutions using equivariant kernels, and characterize the space of such kernels .