Christoph Schnörr University of Heidelberg The assignment flow is a dynamical system that evolves on an elementary statistical manifold and performs image labeling, i.e.~context-sensitive image classification. It provides a smooth and computationally efficient alternative to non-smooth discrete graphical models and enables to study basic problems related to the design and understanding of larger compositional systems for image analysis: supervised labeling, unsupervised label learning and representation learning using optimal control. The talk reports the mathematical ingredients (information theory, discrete optimal transport, geometric integration), recent results and perspectives.