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Ground Metric Learning on Graphs

Matthieu Heitz 1, 2 Nicolas Bonneel 1, 3, 2 David Coeurjolly 1, 2 Marco Cuturi 4 Gabriel Peyré 5
1 M2DisCo - Geometry Processing and Constrained Optimization
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
2 Origami - Origami
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
3 GeoMod - Modélisation Géométrique, Géométrie Algorithmique, Fractales
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
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. The challenge of selecting it adaptively and algorithmically from prior knowledge, the so-called ground metric learning (GML) problem, has therefore appeared in various settings. In this paper, we consider the GML problem 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 the evolution of the color palette induced by the modi- fication of lighting and materials.
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Preprints, Working Papers, ...
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Contributor : David Coeurjolly <>
Submitted on : Saturday, November 16, 2019 - 10:41:47 AM
Last modification on : Thursday, November 5, 2020 - 12:58:08 PM

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  • HAL Id : hal-02366636, version 1
  • ARXIV : 1911.03117


Matthieu Heitz, Nicolas Bonneel, David Coeurjolly, Marco Cuturi, Gabriel Peyré. Ground Metric Learning on Graphs. 2019. ⟨hal-02366636⟩



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