用规范协变理论分析深度神经网络的稳定性与宽度效应。
Gauge-covariant stochastic neural fields: Stability and finite-width effects
- 构建含复数场与随机深度变量的场论模型。
- 发现有限宽度导致谱形变，但临界条件不变。
- 适合研究神经网络动力学与混沌边缘的学者。
我们发展了一种规范协变的随机有效场论，用于分析深度神经系统的稳定性和有限宽度效应。该模型采用经典对易场：一个复数物质场、一个实阿贝尔规范场和一个虚构的随机深度变量。利用Martin-Siggia-Rose-Janssen-de Dominicis形式化方法，推导出其泛函表示，并建立双副本线性响应构造，定义最大李雅普诺夫指数和混沌边缘的放大因子。有限宽度效应表现为修饰核函数的微扰修正，且在固定核几何下，边际性条件在所考虑阶数内保持不变。数值上，有限宽度的多层感知机遵循均值场不稳定性阈值，而线性随机有效部分可重现预测的低频谱形变。
原文摘要 · Abstract (English)
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.
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