Universal Approximation Theorem for Equivariant Maps by Group CNNs
This work is significant for researchers in machine learning and deep learning who are interested in the theoretical foundations of equivariant neural networks, particularly for complex data structures and symmetries.
This paper tackles the problem of proving universal approximation theorems for equivariant maps using Group CNNs. It provides a unified method that can handle non-linear equivariant maps between infinite-dimensional spaces for non-compact groups, a significant advantage over prior work.
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.