## Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS

### Lin Chen · Sheng Xu

Keywords: [ Laplace kernel ] [ Reproducing kernel Hilbert space ] [ Neural tangent kernel ] [ Singularity analysis ]

Abstract: We prove that the reproducing kernel Hilbert spaces (RKHS) of a deep neural tangent kernel and the Laplace kernel include the same set of functions, when both kernels are restricted to the sphere $\mathbb{S}^{d-1}$. Additionally, we prove that the exponential power kernel with a smaller power (making the kernel less smooth) leads to a larger RKHS, when it is restricted to the sphere $\mathbb{S}^{d-1}$ and when it is defined on the entire $\mathbb{R}^d$.