Linear CNNs Discover the Statistical Structure of the Dataset Using Only the Most Dominant Frequencies

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. In addition to a neural network's pre-defined architecture, the parameters of the network obtain an implicit structure during training. For example, it has been shown that weight matrices can exhibit structural patterns, such as clusters and branches (Voss et al., 2021; Casper et al., 2022) . On the other hand, the input dataset also has an implicit structure arising from patterns and relationships between the samples. E.g., in a classification task, dogs are more visually similar