State-of-the-art neural networks are vulnerable to adversarial examples; they can easily misclassify inputs that are imperceptibly different than their training and test data. In this work, we establish that the use of cross-entropy loss function and the low-rank features of the training data have responsibility for the existence of these inputs. Based on this observation, we suggest that addressing adversarial examples requires rethinking the use of cross-entropy loss function and looking for an alternative that is more suited for minimization with low-rank features. In this direction, we present a training scheme called differential training, which uses a loss function defined on the differences between the features of points from opposite classes. We show that differential training can ensure a large margin between the decision boundary of the neural network and the points in the training dataset. This larger margin increases the amount of perturbation needed to flip the prediction of the classifier and makes it harder to find an adversarial example with small perturbations. We test differential training on a binary classification task with CIFAR-10 dataset and demonstrate that it radically reduces the ratio of images for which an adversarial example could be found -- not only in the training dataset, but in the test dataset as well.
 We present a method for training a similarity metric from data. The method can be used for recognition or verification applications where the number of categories is very large and not known during training, and where the number of training samples for a single category is very small. The idea is to learn a function that maps input patterns into a target space such that the L sub 1 norm in the target space approximates the "semantic" distance in the input space. The method is applied to a face verification task. The learning process minimizes a discriminative loss function that drives the similarity metric to be small for pairs of faces from the same person, and large for pairs from different persons. The mapping from raw to the target space is a convolutional network whose architecture is designed for robustness to geometric distortions. The system is tested on the Purdue AR face database which has a very high degree of variability in the pose, lighting, expression, position, and artificial occlusions such as dark glasses and obscuring scarves. This paper describes the development of an algorithm for verification of signatures written on a touch-sensitive pad. The signature verification algorithm is based on an artificial neural network. The novel network presented here, called a “Siamese” time delay neural network, consists of two identical networks joined at their output. During training the network learns to measure the similarity between pairs of signatures. When used for verification, only one half of the Siamese network is evaluated. The output of this half network is the feature vector for the input signature. Verification consists of comparing this feature vector with a stored feature vector for the signer. Signatures closer than a chosen threshold to this stored representation are accepted, all other signatures are rejected as forgeries. System performance is illustrated with experiments performed in the laboratory.