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.

Capsule networks are constrained by their, relative, inability to deeper in a parameter-inexpensive manner, and also by the general lack of equivariance guarantees. As a step towards bridging these two gaps, we present a new variation of capsule networks termed Space-of-Variation networks (SOVNET). Each layer in SOVNET learns to projectively represent the manifold of legal pose variations for a set of capsules, using learnable neural network - one per capsule-type. Thus, shallower capsules from a local pool predict a deeper capsule by being input into the neural network associated with the type of deeper capsule. In order to capture local object-structures better, benefit from increased parameter-sharing, and have equivariance guarantees; group-equivariant convolutions are used in the prediction mechanism. Further, a new routing algorithm based on the degree-centrality of graph nodes is presented. Experiments on affinely transformed versions of MNIST and FashionMNIST showcase the superiority of SOVNET over certain capsule-network baselines.