揭示随机神经网络输出波动的相变规律，发现三种极限行为
Phase Transitions in the Fluctuations of Functionals of Random Neural Networks
- 通过协方差函数不动点分析网络深度变化时的统计极限
- 随深度增加，输出泛函收敛至高斯场、标准高斯分布或第Q阶维纳混沌分布
- 首次将协方差迭代算子的不动点稳定性作为相变判据，适合概率与深度学习交叉研究者
我们建立了无限宽随机神经网络在d维球面上高斯输出泛函序列的中心极限定理与非中心极限定理。随着网络深度增加，这些泛函的渐近行为关键取决于协方差函数的不动点，导致三种不同极限情形：收敛至极限高斯场的相同泛函、收敛至高斯分布、收敛至第Q阶维纳混沌分布。证明中运用了经典工具（厄米展开、图公式、Stein-Malliavin方法），也引入了全新思路：渐近行为由与协方差相关的迭代算子的不动点结构决定，其性质与稳定性决定了不同极限相的出现。
原文摘要 · Abstract (English)
We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere . We show that the asymptotic behaviour of these functionals as the depth of the network increases depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes: convergence to the same functional of a limiting Gaussian field, convergence to a Gaussian distribution, convergence to a distribution in the Qth Wiener chaos. Our proofs exploit tools that are now classical (Hermite expansions, Diagram Formula, Stein-Malliavin techniques), but also ideas which have never been used in similar contexts: in particular, the asymptotic behaviour is determined by the fixed-point structure of the iterative operator associated with the covariance, whose nature and stability governs the different limiting regimes.
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