Characterization and Design of Generalized Convolutional Neural Network

The group convolution and representation theory give a strong support for generalized convolutional neural network. The generalized convolutional neural network (G-CNN) has been applied to learning problems and achieved the state-of-art performance. But a theoretical support for details of network architecture design is still lacking. In this work, we first analyze the necessary and sufficient condition for a neural network to be group equivariant when the group acts on the sub-domain of input/output. We then analyze the multiple equivariance case. After that, we show that the generalized convolution mapping to a quotient space is a projection of the image of a generalized convolution which maps to the maximum quotient space. This can be used to obtain guidelines for choosing the feature size of hidden layer.