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Under mild conditions on the network initialization we derive a power seriesexpansion for the Neural Tangent Kernel (NTK) of arbitrarily deep feedforwardnetworks in the infinite width limit. We provide expressions for thecoefficients of this power series which depend on both the Hermite coefficientsof the activation function as well as the depth of the network. We observefaster decay of the Hermite coefficients leads to faster decay in the NTKcoefficients and explore the role of depth. Using this series, first we relatethe effective rank of the NTK to the effective rank of the input-data Gram.Second, for data drawn uniformly on the sphere we study the eigenvalues of theNTK, analyzing the impact of the choice of activation function. Finally, forgeneric data and activation functions with sufficiently fast Hermitecoefficient decay, we derive an asymptotic upper bound on the spectrum of theNTK.

Characterizing the Spectrum of the NTK via a Power Series Expansion