On the regularization of convolutional kernel tensors in neural networks

Convolutional neural network is an important model in deep learning, where a convolution operation can be represented by a tensor. To avoid exploding/vanishing gradient problems and to improve the generalizability of a neural network, it is desirable to have a convolution operation that nearly preserves the norm, or to have the singular values of the transformation matrix corresponding to the tensor bounded around 1. We propose a penalty function that can constrain the singular values of the transformation matrix to be around 1. We derive an algorithm to carry out the gradient descent minimization of this penalty function in terms of convolution kernel tensors. Numerical examples are presented to demonstrate the effectiveness of the method.