$C^*$-Algebraic Machine Learning: Moving in a New Direction
This is an incremental proposal for researchers in machine learning and mathematics, aiming to introduce a new theoretical framework without demonstrated practical impact.
The paper tackles the problem of unifying and extending machine learning methods by proposing a new direction called $C^*$-algebraic ML, which uses $C^*$-algebras to generalize existing strategies and create a framework for more diverse data models, though no concrete results or numbers are provided.
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.