  Recent empirical work has shown that hierarchical convolutional kernels
inspired by convolutional neural networks (CNNs) significantly improve the
performance of kernel methods in image classification tasks. A widely accepted
explanation for their success is that these architectures encode hypothesis
classes that are suitable for natural images. However, understanding the
precise interplay between approximation and generalization in convolutional
architectures remains a challenge. In this paper, we consider the stylized
setting of covariates (image pixels) uniformly distributed on the hypercube,
and characterize exactly the RKHS of kernels composed of single layers of
convolution, pooling, and downsampling operations. We use this characterization
to compute sharp asymptotics of the generalization error for any given function
in high-dimension. In particular, we quantify the gain in sample complexity
brought by enforcing locality with the convolution operation and approximate
translation invariance with average pooling. Notably, these results provide a
precise description of how convolution and pooling operations trade off
approximation with generalization power in one layer convolutional kernels.
