@inproceedings{
sonoda2023joint,
title={Joint Group Invariant Functions on Data-Parameter Domain Induce Universal Neural Networks},
author={Sho Sonoda and Hideyuki Ishi and Isao Ishikawa and Masahiro Ikeda},
booktitle={NeurIPS 2023 Workshop on Symmetry and Geometry in Neural Representations},
year={2023},
url={https://openreview.net/forum?id=uOjSFxFz5k}
}
The symmetry and geometry of input data are considered to be encoded in the internal data representation inside the neural network, but the specific encoding rule has been less investigated. In this study, we present a systematic method to induce a generalized neural network and its right inverse operator, called the ridgelet transform, from a joint group invariant function on the data-parameter domain. Since the ridgelet transform is an inverse, (1) it can describe the arrangement of parameters for the network to represent a target function, which is understood as the encoding rule, and (2) it implies the universality of the network. Based on the group representation theory, we present a new simple proof of the universality by using Schur's lemma in a unified manner covering a wide class of networks, for example, the original ridgelet transform, formal deep networks, and the dual voice transform. Since traditional universality theorems were demonstrated based on functional analysis, this study sheds light on the group theoretic aspect of the approximation theory, connecting geometric deep learning to abstract harmonic analysis.
Joint Group Invariant Functions on Data-Parameter Domain Induce Universal Neural Networks
Sho Sonoda Hideyuki Ishi Isao Ishikawa Masahiro Ikeda