Ground Metric Learning on Graphs

TL;DR

This paper introduces a method for learning graph-based ground metrics for optimal transport distances, enabling efficient modeling of evolving densities in natural phenomena.

Contribution

It proposes a novel approach to ground metric learning constrained to graph geodesic distances, improving efficiency and applicability in modeling dynamic density evolutions.

Findings

Efficient learning of graph-based ground metrics for OT.

Successful modeling of density evolution in natural phenomena.

Enhanced interpretability of the learned metrics.

Abstract

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.