Universality of Group Convolutional Neural Networks Based on Ridgelet Analysis on Groups

TL;DR

This paper proves the universality of depth-2 group convolutional neural networks (GCNNs) using ridgelet theory, providing a unified and constructive analysis of their function approximation capabilities.

Contribution

It introduces a versatile depth-2 GCNN formulation and derives the first ridgelet transform for GCNNs, enabling systematic and unified universality proofs.

Findings

First ridgelet transform for GCNNs derived

Constructive proof of GCNN universality provided

Applicable to various GCNN architectures like cyclic and E(n)-equivariant networks

Abstract

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 $S[\gamma]=f$. The proposed GCNN covers typical GCNNs such as the cyclic convolution on multi-channel images, networks on permutation-invariant inputs (Deep Sets), and $\mathrm{E}(n)$-equivariant networks. The closed-form expression of the ridgelet transform can describe how the network parameters are organized to represent a function. While it has been known only for fully-connected networks, this study is the first to obtain the ridgelet transform for GCNNs. By discretizing the closed-form expression, we can systematically generate a constructive proof of the $cc$-universality of finite GCNNs. In other words, our universality proofs are more unified and constructive than previous proofs.