Harmonic Decompositions of Convolutional Networks

We present a description of the function space and the smoothness class associated with a convolutional network using the machinery of reproducing kernel Hilbert spaces. We show that the mapping associated with a convolutional network expands into a sum involving elementary functions akin to spherical harmonics. The functional decomposition can be related to functional ANOVA decompositions in nonparametric statistics. Building off this functional characterization, we obtain statistical bounds which highlight an interesting trade-off between the approximation error and the estimation error.