  We present group equivariant capsule networks, a framework to introduce
guaranteed equivariance and invariance properties to the capsule network idea.
Our work can be divided into two contributions. First, we present a generic
routing by agreement algorithm defined on elements of a group and prove that
equivariance of output pose vectors, as well as invariance of output
activations, hold under certain conditions. Second, we connect the resulting
equivariant capsule networks with work from the field of group convolutional
networks. Through this connection, we provide intuitions of how both methods
relate and are able to combine the strengths of both approaches in one deep
neural network architecture. The resulting framework allows sparse evaluation
of the group convolution operator, provides control over specific equivariance
and invariance properties, and can use routing by agreement instead of pooling
operations. In addition, it is able to provide interpretable and equivariant
representation vectors as output capsules, which disentangle evidence of object
existence from its pose.
