  We here present a stepping stone towards a deeper understanding of
convolutional neural networks (CNNs) in the form of a theory of learning in
linear CNNs. Through analyzing the gradient descent equations, we discover that
the evolution of the network during training is determined by the interplay
between the dataset structure and the convolutional network structure. We show
that linear CNNs discover the statistical structure of the dataset with
non-linear, ordered, stage-like transitions, and that the speed of discovery
changes depending on the relationship between the dataset and the convolutional
network structure. Moreover, we find that this interplay lies at the heart of
what we call the ``dominant frequency bias'', where linear CNNs arrive at these
discoveries using only the dominant frequencies of the different structural
parts present in the dataset. We furthermore provide experiments that show how
our theory relates to deep, non-linear CNNs used in practice. Our findings shed
new light on the inner working of CNNs, and can help explain their shortcut
learning and their tendency to rely on texture instead of shape.
