Universality of group convolutional neural networks based on ridgelet analysis on groups
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We investigate the approximation property of group convolutional neural networks (GCNNs) based on the ridgelet theory. We regard a group convolution as a matrix element of a group representation, and formulate a versatile GCNN as a nonlinear mapping between group representations, which covers typical GCNN literatures such as a cyclic convolution on a multi-channel image, permutation-invariant datasets (Deep Sets), and E(n)-equivariant convolutions. The ridgelet transform is an analysis operator of a depth-2 network, namely, it maps an arbitrary given target function f to the weight γ of a network S[γ] so that the network represents the function as S[γ]=f. It has been known only for fully-connected networks, and this study is the first to present the ridgelet transform for (G)CNNs. Since the ridgelet transform is given as a closed-form integral operator, it provides a constructive proof of the cc-universality of GCNNs. Unlike previous universality arguments on CNNs, we do not need to convert/modify the networks into other universal approximators such as invariant polynomials and fully-connected networks.
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