Universal Approximation Theorem for Equivariant Maps by Group CNNs

TL;DR

This paper establishes a unified framework demonstrating that CNNs can universally approximate equivariant maps across various groups, including complex non-linear cases in infinite-dimensional spaces, enhancing their theoretical understanding.

Contribution

It provides a unified method for proving universal approximation theorems for equivariant CNNs across different groups and settings, including non-compact groups and infinite-dimensional spaces.

Findings

Universal approximation theorems for equivariant CNNs are established.

The method applies to non-linear equivariant maps in infinite-dimensional spaces.

Handles non-compact groups, broadening applicability.

Abstract

Group symmetry is inherent in a wide variety of data distributions. Data processing that preserves symmetry is described as an equivariant map and often effective in achieving high performance. Convolutional neural networks (CNNs) have been known as models with equivariance and shown to approximate equivariant maps for some specific groups. However, universal approximation theorems for CNNs have been separately derived with individual techniques according to each group and setting. This paper provides a unified method to obtain universal approximation theorems for equivariant maps by CNNs in various settings. As its significant advantage, we can handle non-linear equivariant maps between infinite-dimensional spaces for non-compact groups.