C^*-Algebraic Machine Learning: Moving in a New Direction

Machine learning has a long collaborative tradition with several fields of mathematics, such as statistics, probability and linear algebra. We propose a new direction for machine learning research: C^*-algebraic ML - a cross-fertilization between C^*-algebra and machine learning. The mathematical concept of C^*-algebra is a natural generalization of the space of complex numbers. It enables us to unify existing learning strategies, and construct a new framework for more diverse and information-rich data models. We explain why and how to use C^*-algebras in machine learning, and provide technical considerations that go into the design of C^*-algebraic learning models in the contexts of kernel methods and neural networks. Furthermore, we discuss open questions and challenges in C^*-algebraic ML and give our thoughts for future development and applications.