Double framed moduli spaces of quiver representations

TL;DR

This paper explores the connection between neural networks and moduli spaces of double framed quiver representations, providing algebraic and geometric descriptions, and showing neural network outputs depend on these moduli.

Contribution

It introduces a novel framework linking neural network behavior to quiver moduli spaces, including a symplectic reduction perspective for ReLU networks.

Findings

Neural network outputs depend only on points in the moduli space.

A new network category is defined linking neural networks to quiver orbits.

ReLU networks can be modeled via symplectic reduction of quiver moduli.

Abstract

Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define a network category whose isomorphism classes of objects correspond to the orbits of quiver representations, in which neural networks map input data. We then prove that the output of a neural network depends only on the corresponding point in the moduli space. Finally, we present a different perspective on mapping neural networks with a specific activation function, called ReLU, to a moduli space using the symplectic reduction approach to quiver moduli.