$C^*$-Algebraic Machine Learning: Moving in a New Direction
Yuka Hashimoto Masahiro Ikeda Hachem Kadri
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.