  The assignment flow recently introduced in the J. Math. Imaging and Vision
58/2 (2017), constitutes a high-dimensional dynamical system that evolves on an
elementary statistical manifold and performs contextual labeling
(classification) of data given in any metric space. Vertices of a given graph
index the data points and define a system of neighborhoods. These neighborhoods
together with nonnegative weight parameters define regularization of the
evolution of label assignments to data points, through geometric averaging
induced by the affine e-connection of information geometry. Regarding
evolutionary game dynamics, the assignment flow may be characterized as a large
system of replicator equations that are coupled by geometric averaging. This
paper establishes conditions on the weight parameters that guarantee
convergence of the continuous-time assignment flow to integral assignments
(labelings), up to a negligible subset of situations that will not be
encountered when working with real data in practice. Furthermore, we classify
attractors of the flow and quantify corresponding basins of attraction. This
provides convergence guarantees for the assignment flow which are extended to
the discrete-time assignment flow that results from applying a
Runge-Kutta-Munthe-Kaas scheme for numerical geometric integration of the
assignment flow. Several counter-examples illustrate that violating the
conditions may entail unfavorable behavior of the assignment flow regarding
contextual data classification.
