Meyer Scetbon, Zaid Harchaoui
We present a description of function spaces and smoothness classes associated with convolutional networks from a reproducing kernel Hilbert space viewpoint. We establish harmonic decompositions of convolutional networks, that is expansions into sums of elementary functions of feature-representation maps implemented by convolutional networks. The elementary functions are related to the spherical harmonics, a fundamental class of special functions on spheres. These harmonic decompositions allow us to characterize the integral operators associated with convolutional networks, and obtain as a result risk bounds for convolutional networks which highlight their behavior in high dimensions.