Quantum Finite Automata and Quiver Algebras

We find an application in quantum finite automata for the ideas and results of [JL21] and [JL22]. We reformulate quantum finite automata with multiple-time measurements using the algebraic notion of near-ring. This gives a unified understanding towards quantum computing and deep learning. When the near-ring comes from a quiver, we have a nice moduli space of computing machines with metric that can be optimized by gradient descent.

Noncommutative Geometry of Computational Models and Uniformization for Framed Quiver Varieties

We formulate a mathematical setup for computational neural networks using noncommutative algebras and near-rings, in motivation of quantum automata. We study the moduli space of the corresponding framed quiver representations, and find moduli of Euclidean and non-compact types in light of uniformization.

Kähler Geometry of Quiver Varieties and Machine Learning

We develop an algebro-geometric formulation for neural networks in machine learning using the moduli space of framed quiver representations. We find natural Hermitian metrics on the universal bundles over the moduli which are compatible with the GIT quotient construction by the general linear group, and show that their Ricci curvatures give a K\"ahler metric on the moduli. Moreover, we use toric moment maps to construct activation functions, and prove the universal approximation theorem for the multi-variable activation function constructed from the complex projective space.