Abstract. A method for learning local affine-covariant regions is presented. We show that maximizing geometric repeatability does not lead to local regions, a.k.a features, that are reliably matched and this necessitates descriptor-based learning. We explore factors that influence such learning and registration: the loss function, descriptor type, geometric parametrization and the trade-off between matchability and geometric accuracy and propose a novel hard negative-constant loss function for learning of affine regions. The affine shape estimator – AffNet – trained with the hard negative-constant loss outperforms the state-of-the-art in bag-of-words image retrieval and wide baseline stereo. The proposed training process does not require precisely geometrically aligned patches. The source codes and trained weights are available at https://github.com/ducha-aiki/affnet