# Ground Metric Learning on Graphs

arXiv: https://arxiv.org/abs/1911.03117 · HTML: https://mlsanity.com/paper/1911.03117/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2019.

## Built on by (most cited first, 17 of 17)

- [Low-Rank Sinkhorn Factorization](https://mlsanity.com/paper/2103.04737/) (2021)
- [Riemannian Metric Learning via Optimal Transport](https://mlsanity.com/paper/2205.09244/) (2022)
- [Low-rank Optimal Transport: Approximation, Statistics and Debiasing](https://mlsanity.com/paper/2205.12365/) (2022)
- [Geodesic Sinkhorn for Fast and Accurate Optimal Transport on Manifolds](https://mlsanity.com/paper/2211.00805/) (2022)
- [Learning to Generate Wasserstein Barycenters](https://mlsanity.com/paper/2102.12178/) (2021)
- [A Heat Diffusion Perspective on Geodesic Preserving Dimensionality Reduction](https://mlsanity.com/paper/2305.19043/) (2023)
- [Joint trajectory and network inference via reference fitting](https://mlsanity.com/paper/2409.06879/) (2024)
- [Unsupervised Ground Metric Learning using Wasserstein Singular Vectors](https://mlsanity.com/paper/2102.06278/) (2021)
- [Gaussian Process regression over discrete probability measures: on the non-stationarity relation between Euclidean and Wasserstein Squared Exponential Kernels](https://mlsanity.com/paper/2212.01310/) (2022)
- [Differentiable Cost-Parameterized Monge Map Estimators](https://mlsanity.com/paper/2406.08399/) (2024)
- [Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization](https://mlsanity.com/paper/2502.07510/) (2025)
- [Fast Marching Energy CNN](https://mlsanity.com/paper/2306.16109/) (2023)
- [Model alignment using inter-modal bridges](https://mlsanity.com/paper/2505.12322/) (2025)
- [Unsupervised Ground Metric Learning](https://mlsanity.com/paper/2507.13094/) (2025)
- [FlowPool: Pooling Graph Representations with Wasserstein Gradient Flows](https://mlsanity.com/paper/2112.09990/) (2021)
- [Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization](https://mlsanity.com/paper/2410.02628/) (2024)
- [Reliable Modeling of Distribution Shifts via Displacement-Reshaped Optimal Transport](https://mlsanity.com/paper/2605.04965/) (2026)

## Abstract (arXiv)

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.