Eigenvalue distribution of the Neural Tangent Kernel in the quadratic scaling

• URL: http://arxiv.org/abs/2508.20036v1
• Date: Wed, 27 Aug 2025 16:41:01 GMT
• Title: Eigenvalue distribution of the Neural Tangent Kernel in the quadratic scaling
• Authors: Lucas Benigni, Elliot Paquette,
• Abstract summary: We compute the eigenvalue distribution of the neural tangent kernel of a two-layer neural network under a specific scaling of dimension.<n>We describe the distribution as the free multiplicative convolution of the Marchenko--Pastur distribution with a deterministic distribution depending on $sigma$ and $D$.
• Score: 5.142160533428576
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We compute the asymptotic eigenvalue distribution of the neural tangent kernel of a two-layer neural network under a specific scaling of dimension. Namely, if $X\in\mathbb{R}^{n\times d}$ is an i.i.d random matrix, $W\in\mathbb{R}^{d\times p}$ is an i.i.d $\mathcal{N}(0,1)$ matrix and $D\in\mathbb{R}^{p\times p}$ is a diagonal matrix with i.i.d bounded entries, we consider the matrix \[ \mathrm{NTK} = \frac{1}{d}XX^\top \odot \frac{1}{p} \sigma'\left( \frac{1}{\sqrt{d}}XW \right)D^2 \sigma'\left( \frac{1}{\sqrt{d}}XW \right)^\top \] where $\sigma'$ is a pseudo-Lipschitz function applied entrywise and under the scaling $\frac{n}{dp}\to \gamma_1$ and $\frac{p}{d}\to \gamma_2$. We describe the asymptotic distribution as the free multiplicative convolution of the Marchenko--Pastur distribution with a deterministic distribution depending on $\sigma$ and $D$.

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