Nonlinear Ranking Loss on Riemannian Potato Embedding

We propose a rank-based metric learning method by leveraging a concept of the Riemannian Potato for better separating non-linear data. By exploring the geometric properties of Riemannian manifolds, the proposed loss function optimizes the measure of dispersion using the distribution of Riemannian distances between a reference sample and neighbors and builds a ranked list according to the similarities. We show the proposed function can learn a hypersphere for each class, preserving the similarity structure inside it on Riemannian manifold. As a result, compared with Euclidean distance-based metric, our method can further jointly reduce the intra-class distances and enlarge the inter-class distances for learned features, consistently outperforming state-of-the-art methods on three widely used non- linear datasets.