Many convolutional neural networks (CNN's) have a feed-forward structure. In this paper, we model a general framework for analyzing the Lipschitz bounds of CNN's and propose a linear program that estimates these bounds. Several CNN's, including the scattering networks, the AlexNet and the GoogleNet, are studied numerically. In these practical numerical examples, estimations of local Lipschitz bounds are compared to these theoretical bounds. Based on the Lipschitz bounds, we next establish concentration inequalities for the output distribution with respect to a stationary random input signal. The Lipschitz bound is further used to perform nonlinear discriminant analysis that measures the separation between features of different classes.