The softmax transformation is a key component of several statistical learning models, encompassing multinomial logistic regression, action selection in reinforcement learning, and neural networks for multi-class classification. Recently, it has also been used to design attention mechanisms in neural networks, with important achievements in machine translation, image caption generation, speech recognition, and various tasks in natural language understanding and computation learning. In this talk, I will describe sparsemax, a new activation function similar to the traditional softmax, but able to output sparse probabilities. After deriving its properties, I will show how its Jacobian can be efficiently computed, enabling its use in a neural network trained with backpropagation. Then, I will propose a new smooth and convex loss function which is the sparsemax analogue of the logistic loss. An unexpected connection between this new loss and the Huber classification loss will be revealed. We obtained promising empirical results in multi-label classification problems and in attention-based neural networks for natural language inference. For the latter, we achieved a similar performance as the traditional softmax, but with a selective, more compact, attention focus.