Given a set of images, our goal is to map each image to a point in a feature space such that, not only point proximity indicates visual similarity, but where it is located directly encodes how prototypical the image is according to the dataset.
Our key insight is to perform unsupervised feature learning in hyperbolic instead of Euclidean space, where the distance between points still reflects image similarity, yet we gain additional capacity for representing prototypicality with the location of the point: The closer it is to the origin, the more prototypical it is. The latter property is simply emergent from optimizing the metric learning objective: The image similar to many training instances is best placed at the center of corresponding points in Euclidean space, but closer to the origin in hyperbolic space.
We propose an unsupervised feature learning algorithm in \underline{H}yperbolic space with sphere p\underline{ACK}ing. HACK first generates uniformly packed particles in the Poincar\'e ball of hyperbolic space and then assigns each image uniquely to a particle. With our feature mapper simply trained to spread out training instances in hyperbolic space, we observe that images move closer to the origin with congealing - a warping process that aligns all the images and makes them appear more common and similar to each other, validating our idea of unsupervised prototypicality discovery. We demonstrate that our data-driven prototypicality provides an easy and superior unsupervised instance selection to reduce sample complexity, increase model generalization with atypical instances and robustness with typical ones.