Gauge-covariant stochastic neural fields: Stability and finite-width effects

TL;DR

This paper introduces a gauge-covariant stochastic effective field theory for deep neural networks, analyzing stability, finite-width effects, and chaos using advanced formalism and numerical validation.

Contribution

It develops a novel gauge-covariant framework incorporating finite-width effects and stability analysis for deep neural systems.

Findings

Finite-width multilayer perceptrons follow the mean-field instability threshold.

A linear stochastic sector reproduces low-frequency spectral deformation.

Finite-width effects appear as perturbative corrections to dressed kernels.

Abstract

We develop a gauge-covariant stochastic effective field theory for stability and finite-width effects in deep neural systems. The model uses classical commuting fields: a complex matter field, a real Abelian connection field, and a fictitious stochastic depth variable. Using the Martin--Siggia--Rose--Janssen--de~Dominicis formalism, we derive its functional representation and a two-replica linear-response construction defining the maximal Lyapunov exponent and the amplification factor for the edge of chaos. Finite-width effects appear as perturbative corrections to dressed kernels, and the marginality condition remains unchanged at the order considered for fixed kernel geometry. Numerically, finite-width multilayer perceptrons follow the mean-field instability threshold, and a linear stochastic effective sector reproduces the predicted low-frequency spectral deformation.