We study the Conjugate Kernel associated to a multi-layer linear-width feed-forward neural network with random weights, biases and data. We show that the empirical spectral distribution of the Conjugate Kernel converges to a deterministic limit. More precisely we obtain a deterministic equivalent for its Stieltjes transform and its resolvent, with quantitative bounds involving both the dimension and the spectral parameter. The limiting equivalent objects are described by iterating free convolution of measures and classical matrix operations involving the parameters of the model.

翻译：我们研究了具有随机权重、偏置和数据的多层线性宽度前馈神经网络的共轭核。我们证明，该共轭核的经验谱分布收敛于一个确定性极限。更精确地，我们获得了其Stieltjes变换和预解式的确定性等价，并给出了涉及维度和谱参数的定量界限。这些极限等价对象通过遍历测度的自由卷积以及涉及模型参数的经典矩阵运算来描述。