  Convolutional neural networks (CNNs) have achieved remarkable performance in
various fields, particularly in the domain of computer vision. However, why
this architecture works well remains to be a mystery. In this work we move a
small step toward understanding the success of CNNs by investigating the
learning dynamics of a two-layer nonlinear convolutional neural network over
some specific data distributions. Rather than the typical Gaussian assumption
for input data distribution, we consider a more realistic setting that each
data point (e.g. image) contains a specific pattern determining its class
label. Within this setting, we both theoretically and empirically show that
some convolutional filters will learn the key patterns in data and the norm of
these filters will dominate during the training process with stochastic
gradient descent. And with any high probability, when the number of iterations
is sufficiently large, the CNN model could obtain 100% accuracy over the
considered data distributions. Our experiments demonstrate that for practical
image classification tasks our findings still hold to some extent.
