  This paper introduces a novel nonlocal partial difference equation (G-PDE)
for labeling metric data on graphs. The G-PDE is derived as nonlocal
reparametrization of the assignment flow approach that was introduced in
\textit{J.~Math.~Imaging \& Vision} 58(2), 2017. Due to this parameterization,
solving the G-PDE numerically is shown to be equivalent to computing the
Riemannian gradient flow with respect to a nonconvex potential. We devise an
entropy-regularized difference-of-convex-functions (DC) decomposition of this
potential and show that the basic geometric Euler scheme for integrating the
assignment flow is equivalent to solving the G-PDE by an established DC
programming scheme. Moreover, the viewpoint of geometric integration reveals a
basic way to exploit higher-order information of the vector field that drives
the assignment flow, in order to devise a novel accelerated DC programming
scheme. A detailed convergence analysis of both numerical schemes is provided
and illustrated by numerical experiments.
