On regularization for a convolutional kernel in neural networks

Convolutional neural network is a very important model of deep learning. It can help avoid the exploding/vanishing gradient problem and improve the generalizability of a neural network if the singular values of the Jacobian of a layer are bounded around $1$ in the training process. We propose a new penalty function for a convolutional kernel to let the singular values of the corresponding transformation matrix are bounded around $1$. We show how to carry out the gradient type methods. The penalty is about the transformation matrix corresponding to a kernel, not directly about the kernel, which is different from results in existing papers. This provides a new regularization method about the weights of convolutional layers. Other penalty functions about a kernel can be devised following this idea in future.