Convolutional neural networks (CNN’s) are widely used in deep learning. The Lipschitz bound is important in the study of stability of CNN’s and computation of the Lipschitz bound is needed for generative networks. We give a general framework for CNN’s and prove that the Lipschitz bound of a CNN can be determined by solving a linear program. We further provide a more explicit expression for a suboptimal bound. The Lipschitz bound is also important in stationary process models and we establish that it can be directly useful for training a discriminative network.