Kähler Geometry of Quiver Varieties and Machine Learning
This work provides a foundational mathematical framework for neural networks, potentially benefiting researchers in machine learning and algebraic geometry by linking these fields.
The authors tackled the problem of formulating neural networks using algebraic geometry by developing an algebro-geometric framework based on moduli spaces of framed quiver representations, resulting in the construction of Kähler metrics and activation functions, and proving the universal approximation theorem for multi-variable activation functions derived from complex projective spaces.
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 Kähler 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.