Our theoretical understanding of the inner workings of general convolutional
neural networks (CNN) is limited. We here present a new stepping stone towards
such understanding in the form of a theory of learning in linear CNNs. By
analyzing the gradient descent equations, we discover that using convolutions
leads to a mismatch between the dataset structure and the network structure. We
show that linear CNNs discover the statistical structure of the dataset with
non-linear, stage-like transitions, and that the speed of discovery changes
depending on this structural mismatch. Moreover, we find that the mismatch 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. Our findings can help
explain several characteristics of general CNNs, such as their shortcut
learning and their tendency to rely on texture instead of shape.

这篇文章介绍了线性卷积神经网络学习的理论，发现卷积操作会导致数据集结构和网络结构不匹配，而线性卷积神经网络则通过非线性阶段性的转变发现数据集的统计结构，这种结构不匹配是影响神经网络性能的核心问题之一，同时还解释了卷积神经网络偏向于纹理而非形状的现象。