Adversarially robust neural network decision boundaries via tropical geometry

We introduce a simple, easy to implement, and computationally efficient tropical convolutional neural network architecture that is robust against adversarial attacks. We exploit the tropical nature of piece-wise linear neural networks by embedding the data in the tropical projective torus. This can be accomplished with a single additional hidden layer called a tropical embedding layer, and can in principle be added to any neural network architecture. We study the geometry of the resulting decision boundary, and find that like adversarial training and various regularization techniques that have been proposed, adding the tropical embedding layer tends to increase the number of linear regions associated with the decision boundaries. Our numerical experiments show that our approach achieves state-of-the-art levels of adversarial robustness, while requiring much less computational time than adversarial training.