  This paper introduces the unsupervised assignment flow that couples the
assignment flow for supervised image labeling with Riemannian gradient flows
for label evolution on feature manifolds. The latter component of the approach
encompasses extensions of state-of-the-art clustering approaches to
manifold-valued data. Coupling label evolution with the spatially regularized
assignment flow induces a sparsifying effect that enables to learn compact
label dictionaries in an unsupervised manner. Our approach alleviates the
requirement for supervised labeling to have proper labels at hand, because an
initial set of labels can evolve and adapt to better values while being
assigned to given data. The separation between feature and assignment manifolds
enables the flexible application which is demonstrated for three scenarios with
manifold-valued features. Experiments demonstrate a beneficial effect in both
directions: adaptivity of labels improves image labeling, and steering label
evolution by spatially regularized assignments leads to proper labels, because
the assignment flow for supervised labeling is exactly used without any
approximation for label learning.
