VC dimensions of group convolutional neural networks

Due to impressive results in image recognition, convolutional neural networks (CNNs) have become one of the most widely-used neural network architectures [12, 13]. It is believed that one of the main reasons for the efficiency of CNNs is their ability to convert translation symmetry of the data into a built-in translationequivariance property of the neural network without exhausting the data to learn the equivariance [4, 15]. Based on this intuition, other data symmetries have recently been incorporated into neural network architectures. Group convolutional neural networks (G-CNNs) are a natural generalization of CNNs that can be equivariant with respect to rotation [5, 24, 23, 9], scale [21, 20, 1], and other symmetries defined by matrix groups [7]. Moreover, every neural network that is equivariant to the action of a group on its input is a G-CNN, where the convolutions are with respect to the group, [11] (see Theorem 2.10 below).