Theory of deep convolutional neural networks II: Spherical analysis

Ding-Xuan Zhou

Abstract:

powerful in many practical applications, but it lacks enough theoretical verifications. In this paper,
we consider a family of deep convolutional neural networks applied to approximate functions on the
unit sphere Sd−1 of Rd. Our analysis presents rates of uniform approximation when the approximated
function lies in the Sobolev space Wr∞
(Sd−1) with r > 0 or takes an additive ridge form. Our work
verifies theoretically the modelling and approximation ability of deep convolutional neural networks
followed by downsampling and one fully connected layer or two. The key idea of our spherical analysis
is to use the inner product form of the reproducing kernels of the spaces of spherical harmonics and
then to apply convolutional factorizations of filters to realize the generated linear features.