Harmonic Decompositions of Convolutional Networks

We consider 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 increasing order. The elementary functions are related to the spherical harmonics, a fundamental class of special functions on spheres. The harmonic decompositions allow us to characterize the integral operators associated with convolutional networks, and obtain as a result statistical bounds for convolutional networks.