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.

在这项工作中，我们建立了最佳控制与训练深卷卷积神经网络（CNN）之间的关系。我们表明，CNN中的正向传播可以被解释为时间依赖性的非线性微分方程，并学习为控制微分方程的参数，以便网络近似于给定培训数据的数据标签关系。使用这种连续的解释，我们得出了两种新方法来扩展CNN相对于两个不同的维度。第一类多尺度方法通过延长和限制CNN参数连接低分辨率和高分辨率数据。我们证明，这可以使用经过低分辨率图像训练的CNN对高分辨率图像进行分类，反之亦然，并启动了学习过程。第二类的多尺度方法将浅网络和深网连接起来，并导致新的培训策略，这些培训策略逐渐增加了CNN的深度，同时重新使用初始化参数。