  Optimal transport (OT) distances between probability distributions are
parameterized by the ground metric they use between observations. Their
relevance for real-life applications strongly hinges on whether that ground
metric parameter is suitably chosen. Selecting it adaptively and
algorithmically from prior knowledge, the so-called ground metric learning GML)
problem, has therefore appeared in various settings. We consider it in this
paper when the learned metric is constrained to be a geodesic distance on a
graph that supports the measures of interest. This imposes a rich structure for
candidate metrics, but also enables far more efficient learning procedures when
compared to a direct optimization over the space of all metric matrices. We use
this setting to tackle an inverse problem stemming from the observation of a
density evolving with time: we seek a graph ground metric such that the OT
interpolation between the starting and ending densities that result from that
ground metric agrees with the observed evolution. This OT dynamic framework is
relevant to model natural phenomena exhibiting displacements of mass, such as
for instance the evolution of the color palette induced by the modification of
lighting and materials.
