  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.
