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.

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**Caption**: *Metric learning example in RGB spage. 1st row: ground truth, the country1 dataset. Rows 2-4: color transfer of each interpolated histograms in Figure 16. When comparing with the ground truth, we see that our method recreates sunset-like colors, as opposed to the two other methods. Row 5: direct transfer of each frame of the seldovia2 dataset on the first frame of the country1 dataset. Our method is able to preserve the original colors of the day image, contrary to a direct transfer.*

```
@techreport{heitz19GroundMetric,
archiveprefix = {arXiv},
author = {Matthieu Heitz and Nicolas Bonneel and David Coeurjolly and Marco Cuturi and Gabriel Peyré},
eprint = {1911.03117},
institution = {arXiv},
month = {November},
number = {1911.03117},
primaryclass = {stat.ML},
title = {Ground Metric Learning on Graphs},
url = {https://arxiv.org/abs/1911.03117},
year = {2019}
}
```