Learning across scales - A multiscale method for Convolution Neural Networks

In this work we establish the relation between optimal control and training deep Convolution Neural Networks (CNNs). We show that the forward propagation in CNNs can be interpreted as a time-dependent nonlinear differential equation and learning as controlling the parameters of the differential equation such that the network approximates the data-label relation for given training data. Using this continuous interpretation we derive two new methods to scale CNNs with respect to two different dimensions. The first class of multiscale methods connects low-resolution and high-resolution data through prolongation and restriction of CNN parameters. We demonstrate that this enables classifying high-resolution images using CNNs trained with low-resolution images and vice versa and warm-starting the learning process. The second class of multiscale methods connects shallow and deep networks and leads to new training strategies that gradually increase the depths of the CNN while re-using parameters for initializations.