  The assignment flow is a smooth dynamical system that evolves on an
elementary statistical manifold and performs contextual data labeling on a
graph. We derive and introduce the linear assignment flow that evolves
nonlinearly on the manifold, but is governed by a linear ODE on the tangent
space. Various numerical schemes adapted to the mathematical structure of these
two models are designed and studied, for the geometric numerical integration of
both flows: embedded Runge-Kutta-Munthe-Kaas schemes for the nonlinear flow,
adaptive Runge-Kutta schemes and exponential integrators for the linear flow.
All algorithms are parameter free, except for setting a tolerance value that
specifies adaptive step size selection by monitoring the local integration
error, or fixing the dimension of the Krylov subspace approximation. These
algorithms provide a basis for applying the assignment flow to machine learning
scenarios beyond supervised labeling, including unsupervised labeling and
learning from controlled assignment flows.
