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In this paper, we provide a strategy to determine the eigenvalue decay rate(EDR) of a large class of kernel functions defined on a general domain ratherthan $\mathbb S^{d}$. This class of kernel functions include but are notlimited to the neural tangent kernel associated with neural networks withdifferent depths and various activation functions. After proving that thedynamics of training the wide neural networks uniformly approximated that ofthe neural tangent kernel regression on general domains, we can furtherillustrate the minimax optimality of the wide neural network provided that theunderground truth function $f\in [\mathcal H_{\mathrm{NTK}}]^{s}$, aninterpolation space associated with the RKHS $\mathcal{H}_{\mathrm{NTK}}$ ofNTK. We also showed that the overfitted neural network can not generalize well.We believe our approach for determining the EDR of kernels might be also ofindependent interests.

On the Eigenvalue Decay Rates of a Class of Neural-Network Related Kernel Functions Defined on General Domains