Title: LEARNING ADVERSARIAL EXAMPLES WITH RIEMANNIAN GEOMETRY. Abstract: Adversarial examples, referred to as augmented data points generated by imperceptible perturbation of input samples, have recently drawn much attention. Well-crafted adversarial examples may even mislead state-of-the-art deep models to make wrong predictions easily. To alleviate this problem, many studies focus on investigating how adversarial examples can be generated and/or resisted. All the existing work handles this problem in the Euclidean space, which may however be unable to describe data geometry. In this paper, we propose a generalized framework that addresses the learning problem of adversarial examples with Riemannian geometry. Specifically, we define the local coordinate systems on Riemannian manifold, develop a novel model called Adversarial Training with Riemannian Manifold, and design a series of theory that manages to learn the adversarial examples in the Riemannian space feasibly and efficiently. The proposed work is important in that (1) it is a generalized learning methodology since Riemmanian manifold space would be degraded to the Euclidean space in a special case; (2) it is the first work to tackle the adversarial example problem tractably through the perspective of geometry; (3) from the perspective of geometry, our method leads to the steepest direction of the loss function. We also provide a series of theory showing that our proposed method can truly find the decent direction for the loss function with a comparable computational time against traditional adversarial methods. Finally, the proposed framework demonstrates superior performance to the traditional counterpart methods on benchmark data including MNIST, CIFAR-10 and SVHN.