This paper proposes a framework for learning features that are robust to data variation, which is particularly important when only a limited number of trainingsamples are available. The framework makes it possible to tradeoff the discriminative value of learned features against the generalization error of the learning algorithm. Robustness is achieved by encouraging the transform that maps data to features to be a local isometry. This geometric property is shown to improve (K, \epsilon)-robustness, thereby providing theoretical justification for reductions in generalization error observed in experiments. The proposed optimization frameworkis used to train standard learning algorithms such as deep neural networks. Experimental results obtained on benchmark datasets, such as labeled faces in the wild,demonstrate the value of being able to balance discrimination and robustness.