Log In Sign Up

Spectral Analysis of the Neural Tangent Kernel for Deep Residual Networks

by   Yuval Belfer, et al.

Deep residual network architectures have been shown to achieve superior accuracy over classical feed-forward networks, yet their success is still not fully understood. Focusing on massively over-parameterized, fully connected residual networks with ReLU activation through their respective neural tangent kernels (ResNTK), we provide here a spectral analysis of these kernels. Specifically, we show that, much like NTK for fully connected networks (FC-NTK), for input distributed uniformly on the hypersphere 𝕊^d-1, the eigenfunctions of ResNTK are the spherical harmonics and the eigenvalues decay polynomially with frequency k as k^-d. These in turn imply that the set of functions in their Reproducing Kernel Hilbert Space are identical to those of FC-NTK, and consequently also to those of the Laplace kernel. We further show, by drawing on the analogy to the Laplace kernel, that depending on the choice of a hyper-parameter that balances between the skip and residual connections ResNTK can either become spiky with depth, as with FC-NTK, or maintain a stable shape.


page 1

page 2

page 3

page 4


On the Similarity between the Laplace and Neural Tangent Kernels

Recent theoretical work has shown that massively overparameterized neura...

A Kernel Perspective of Skip Connections in Convolutional Networks

Over-parameterized residual networks (ResNets) are amongst the most succ...

On the Spectral Bias of Convolutional Neural Tangent and Gaussian Process Kernels

We study the properties of various over-parametrized convolutional neura...

Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS

We prove that the reproducing kernel Hilbert spaces (RKHS) of a deep neu...

On the Power of Shallow Learning

A deluge of recent work has explored equivalences between wide neural ne...

Uniform Generalization Bounds for Overparameterized Neural Networks

An interesting observation in artificial neural networks is their favora...

A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels

Group equivariant convolutional networks (GCNNs) endow classical convolu...