Deep Ridgelet Transform: Voice with Koopman Operator Proves Universality of Formal Deep Networks

TL;DR

This paper presents a theoretical framework linking deep neural networks to group actions and the Koopman operator, providing a formal proof of their universality using group theory and the deep ridgelet transform.

Contribution

It introduces a novel formalism representing deep networks as dual voice transforms related to the Koopman operator, offering a new proof of universality based on group theoretic principles.

Findings

Deep neural networks can be modeled as dual voice transforms.

The universality of DNNs is proven using group actions and Schur's lemma.

A formal deep network framework is established through the deep ridgelet transform.

Abstract

We identify hidden layers inside a deep neural network (DNN) with group actions on the data domain, and formulate a formal deep network as a dual voice transform with respect to the Koopman operator, a linear representation of the group action. Based on the group theoretic arguments, particularly by using Schur's lemma, we show a simple proof of the universality of DNNs.