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.

翻译：机器学习与数学的多个领域（如统计学、概率论和线性代数）长期保持合作关系。我们提出机器学习研究的新方向：$C^*$-代数机器学习——$C^*$-代数与机器学习的交叉融合。$C^*$-代数的数学概念是复数空间的自然推广，使我们能够统一现有学习策略，并构建更丰富信息数据模型的新框架。我们阐释了在机器学习中使用$C^*$-代数的原因与方法，并提供了在核方法和神经网络背景下设计$C^*$-代数学习模型的技术考量。此外，我们讨论了$C^*$-代数机器学习中的开放问题与挑战，并对未来发展与潜在应用提出思考。