Noncommutative geometry for computing machines
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Abstract
Motivated by the quantum measurement problem, we develop an algebro-geometric formulation using noncommutative algebras and near-rings for the computational model of quantum finite automata. For quiver near-rings, it can be interpreted as an artificial neural network. We find a natural way to make the model well-defined over the moduli space of framed quiver representations, in which Hermitian metrics on the universal bundles play a crucial role. We construct a Hermitian metric that admits an explicit expression merely in terms of the quiver algebra and prove that the Ricci curvature gives a Kaehler metric in the case of acyclic quivers. Motivated from uniformization, we also construct moduli spaces of Euclidean and non-compact types. In the fiber direction, we find a connection between activation functions and toric geometry. In particular, we prove the universal approximation theorem for the multi-variable activation function constructed from the complex projective space.
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Attribution 4.0 International