Improve the Robustness and Accuracy of Deep Neural Network with $L_{2,\infty}$ Normalization

In this paper, the robustness and accuracy of the deep neural network (DNN) was enhanced by introducing the $L_{2,\infty}$ normalization of the weight matrices of the DNN with Relu as the activation function. It is proved that the $L_{2,\infty}$ normalization leads to large dihedral angles between two adjacent faces of the polyhedron graph of the DNN function and hence smoother DNN functions, which reduces over-fitting. A measure is proposed for the robustness of a classification DNN, which is the average radius of the maximal robust spheres with the sample data as centers. A lower bound for the robustness measure is given in terms of the $L_{2,\infty}$ norm. Finally, an upper bound for the Rademacher complexity of DNN with $L_{2,\infty}$ normalization is given. An algorithm is given to train a DNN with the $L_{2,\infty}$ normalization and experimental results are used to show that the $L_{2,\infty}$ normalization is effective to improve the robustness and accuracy.