Mathematical equivalence between statistical mechanics and machine learning theory has been known since the 20th century, and researches based on such equivalence have provided novel methodology in both theoretical physics and statistical learning theory. For example, algebraic approach in statistical mechanics such as operator algebra enables us to analyze phase transition phenomena mathematically. In this paper, for theoretical physicists who are interested in artificial intelligence, we review and prospect algebraic researches in machine learning theory. If a learning machine has hierarchical structure or latent variables, then the random Hamiltonian cannot be expressed by any quadratic perturbation because it has singularities. To study an equilibrium state defined by such a singular random Hamiltonian, algebraic approach is necessary to derive asymptotic form of the free energy and the generalization error. We also introduce the most recent advance, in fact, theoretical foundation for alignment of artificial intelligence is now being constructed based on algebraic learning theory. This paper is devoted to the memory of Professor Huzihiro Araki who is a pioneer founder of algebraic research in both statistical mechanics and quantum field theory.

翻译：自20世纪以来，统计力学与机器学习理论间的数学等价性已为人所知，基于该等价性的研究为理论物理学与统计学习理论提供了新的方法论。例如，统计力学中的算子代数等代数方法，使我们能够从数学上分析相变现象。本文针对对人工智能感兴趣的理论物理学者，回顾并展望机器学习理论中的代数研究。若学习机具有层次结构或隐变量，则随机哈密顿量因存在奇点而无法用任何二次微扰表示。为研究由此类奇异随机哈密顿量定义的平衡态，需要采用代数方法推导自由能与泛化误差的渐近形式。我们亦介绍最新进展——事实上，基于代数学习理论的人工智能对齐理论基础正在构建中。谨以此文纪念统计力学与量子场论代数研究的先驱奠基者——荒木不二洋教授。