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Deterministic equivalent of the Conjugate Kernel matrix associated to Artificial Neural Networks

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arxiv 2306.05850 v1 pith:OZL6EXSZ submitted 2023-06-09 math.PR stat.ML

classification math.PRstat.ML
keywords conjugatedeterministicequivalentkernelassociatedinvolvingmatrixneural
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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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  1. Eigenvalue distribution of the Neural Tangent Kernel in the quadratic scaling

    math.PR 2025-08 conditional novelty 7.0 of 10

    The limiting eigenvalue distribution of the two-layer NTK in the quadratic scaling n/(dp) tends to a Marchenko-Pastur map applied to a deterministic measure depending on the activation and output weights.

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