We show the universality of depth-2 group convolutional neural networks (GCNNs) in a unified and constructive manner based on the ridgelet theory. Despite widespread use in applications, the approximation property of (G)CNNs has not been well investigated. The universality of (G)CNNs has been shown since the late 2010s. Yet, our understanding on how (G)CNNs represent functions is incomplete because the past universality theorems have been shown in a case-by-case manner by manually/carefully assigning the network parameters depending on the variety of convolution layers, and in an indirect manner by converting/modifying the (G)CNNs into other universal approximators such as invariant polynomials and fully-connected networks. In this study, we formulate a versatile depth-2 continuous GCNN S[\gamma] as a nonlinear mapping between group representations, and directly obtain an analysis operator, called the ridgelet trasform, that maps a given function f to the network parameter \gamma so that S[\gamma] f .

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).