We show the universality of depth-2 group convolutional neural networks (GCNNs) inaunified and constructivemanner based ontheridgelet theory.

This paper focuses on the intersection of the above two directions,i.e.

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 $S[\gamma]=f$.

We are very thankful to all reviewers for their time and valuable comments. We agree that "lots of works have used GCNN for different combinatorial optimization We agree that our benchmark is artificial, and using real-world instances would bring value. Such datasets could be collected, e.g. We intend to include those new results in the final version of the paper. We agree that we should discuss references [a-c] in our literature review.

We show the universality of depth-2 group convolutional neural networks (GC-NNs) in a unified and constructive manner based on the ridgelet theory.

We show the universality of depth-2 group convolutional neural networks (GC-NNs) in a unified and constructive manner based on the ridgelet theory.