与人工神经网络相关的共轭核矩阵的确定性等价物
Deterministic equivalent of the Conjugate Kernel matrix associated to Artificial Neural Networks
AI总结：
该研究针对带随机权重、偏置与数据的多层线性宽度前馈神经网络的共轭核，证明其经验谱分布收敛于确定性极限，推导了Stieltjes变换和预解式的定量确定性等价物，刻画了由自由卷积迭代与矩阵运算描述的极限对象。
摘要：
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对应的共轭核（Conjugate Kernel）。我们证明该共轭核的经验谱分布收敛于一个确定性极限。更确切地说，我们得到了其Stieltjes变换和预解式的确定性等价物，给出了同时涉及维度和谱参数的定量界。这些极限等价对象通过测度的自由卷积迭代以及涉及模型参数的经典矩阵运算来描述。
英文摘要：
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.
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