  Assignment flows denote a class of dynamical models for contextual data
labeling (classification) on graphs. We derive a novel parametrization of
assignment flows that reveals how the underlying information geometry induces
two processes for assignment regularization and for gradually enforcing
unambiguous decisions, respectively, that seamlessly interact when solving for
the flow. Our result enables to characterize the dominant part of the
assignment flow as a Riemannian gradient flow with respect to the underlying
information geometry. We consider a continuous-domain formulation of the
corresponding potential and develop a novel algorithm in terms of solving a
sequence of linear elliptic PDEs subject to a simple convex constraint. Our
result provides a basis for addressing learning problems by controlling such
PDEs in future work.
