The application of Deep Learning (DL) for medical diagnosis is often hampered by two problems. First, the amount of training data may be scarce, as it is limited by the number of patients who have acquired the condition to be diagnosed. Second, the training data may be corrupted by various types of noise. Here, we study the problem of brain tumor detection from magnetic resonance spectroscopy (MRS) data, where both types of problems are prominent. To overcome these challenges, we propose a new method for training a deep neural network that distills particularly representative training examples and augments the training data by mixing these samples from one class with those from the same and other classes to create additional training samples. We demonstrate that this technique substantially improves performance, allowing our method to reach human-expert-level accuracy with just a few thousand training examples. Interestingly, the network learns to rely on features of the data that are usually ignored by human experts, suggesting new directions for future research.