Unsupervised Labeling by Geometric and Spatially Regularized Self-assignment

We introduce and study the unsupervised self-assignment flow for labeling image data (euclidean or manifold-valued) without specifying any class prototypes (labels) beforehand, and without alternating between data assignment and prototype evolution, which is common in unsupervised learning. Rather, a single smooth flow evolving on an elementary statistical manifold is geometrically integrated which assigns given data to itself. Specifying the scale of spatial regularization by geometric averaging suffices to induce a low-rank data representation, the emergence of prototypes together with their number, and the data labeling. Connections to the literature on low-rank matrix factorization and on data representations based on discrete optimal mass transport are discussed.