Position: 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.