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

We propose a new generalization of neural networks with noncommutative $C^*$-algebra. An important feature of $C^*$-algebras is their noncommutative structure of products, but the existing $C^*$-algebra net frameworks have only considered commutative $C^*$-algebras. We show that this noncommutative structure of $C^*$-algebras 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 invariant features with respect to group actions. We also show the validity of our framework numerically, which illustrates its potential power.