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

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.