Ground Metric Learning on Graphs
This work addresses the ground metric learning problem for optimal transport in a graph-constrained setting, offering a more efficient approach for applications involving mass displacements, such as in computer vision or physics, but it appears incremental as it builds on existing optimal transport frameworks.
The paper tackles the problem of learning a ground metric for optimal transport distances by constraining it to be a geodesic distance on a graph, enabling more efficient learning procedures. It applies this to an inverse problem where the goal is to find a graph ground metric such that the optimal transport interpolation matches an observed density evolution over time, with applications in modeling phenomena like color palette changes.
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.