Noncommutative $C^*$-algebra Net: Learning Neural Networks with Powerful Product Structure in $C^*$-algebra

TL;DR

This paper introduces a novel neural network framework based on noncommutative $C^*$-algebras, leveraging their rich product structures to enhance learning capabilities and enable simultaneous multi-network learning and equivariance.

Contribution

It presents a new approach to neural network parameter spaces using noncommutative $C^*$-algebras, demonstrating their effectiveness and broad applicability.

Findings

Numerical experiments confirm the framework's validity.

The approach enables learning multiple related networks with interactions.

It effectively learns equivariant features under group actions.

Abstract

We propose a new generalization of neural network parameter spaces with noncommutative $C^*$-algebra, which possesses a rich noncommutative structure of products. We show that this noncommutative structure induces powerful effects in learning neural networks. Our framework has a wide range of applications, such as learning multiple related neural networks simultaneously with interactions and learning equivariant features with respect to group actions. Numerical experiments illustrate the validity of our framework and its potential power.