Approximation and Learning with Deep Convolutional Models: a Kernel Perspective

The empirical success of deep convolutional networks on tasks involving high-dimensional data such as images or audio suggests that they can efficiently approximate certain functions that are well-suited for such tasks. In this paper, we study this through the lens of kernel methods, by considering simple hierarchical kernels with two or three convolution and pooling layers, inspired by convolutional kernel networks. These achieve good empirical performance on standard vision datasets, while providing a simple enough description of the functional space to shed light on their inductive bias. We show that the RKHS consists of additive models of interaction terms between patches, and that its norm encourages structured spatial similarities between these terms through pooling layers. We then provide generalization bounds which illustrate how pooling yields improved sample complexity guarantees when the target function presents such regularities.