# Unsupervised Ground Metric Learning

arXiv: https://arxiv.org/abs/2507.13094 · HTML: https://mlsanity.com/paper/2507.13094/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 60 references that are in the map.
- First posted on arXiv 17 Jul 2025.

## Builds on (most cited first, 19 of 19)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [CosFace: Large Margin Cosine Loss for Deep Face Recognition](https://mlsanity.com/paper/1801.09414/) (2018)
- [SphereFace: Deep Hypersphere Embedding for Face Recognition](https://mlsanity.com/paper/1704.08063/) (2017)
- [Unsupervised Meta-Learning For Few-Shot Image Classification](https://mlsanity.com/paper/1811.11819/) (2018)
- [DiVA: Diverse Visual Feature Aggregation for Deep Metric Learning](https://mlsanity.com/paper/2004.13458/) (2020)
- [Mining on Manifolds: Metric Learning without Labels](https://mlsanity.com/paper/1803.11095/) (2018)
- [Inverse optimal transport](https://mlsanity.com/paper/1905.03950/) (2019)
- [Ground Metric Learning on Graphs](https://mlsanity.com/paper/1911.03117/) (2019)
- [Human Motion Analysis with Deep Metric Learning](https://mlsanity.com/paper/1807.11176/) (2018)
- [Spectral, Probabilistic, and Deep Metric Learning: Tutorial and Survey](https://mlsanity.com/paper/2201.09267/) (2022)
- [The Gene Mover's Distance: Single-cell similarity via Optimal Transport](https://mlsanity.com/paper/2102.01218/) (2021)
- [Unsupervised Ground Metric Learning using Wasserstein Singular Vectors](https://mlsanity.com/paper/2102.06278/) (2021)
- [Normalized Radon Cumulative Distribution Transforms for Invariance and Robustness in Optimal Transport Based Image Classification](https://mlsanity.com/paper/2506.08761/) (2025)
- [Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization](https://mlsanity.com/paper/2502.07510/) (2025)
- [Random Function Iterations for Consistent Stochastic Feasibility](https://mlsanity.com/paper/1808.05426/) (2018)
- [Random Function Iterations for Stochastic Fixed Point Problems](https://mlsanity.com/paper/2007.06479/) (2020)
- [Fast unsupervised ground metric learning with tree-Wasserstein distance](https://mlsanity.com/paper/2411.07432/) (2024)

## Built on by (most cited first, 1 of 1)

- [Generalizations of the Normalized Radon Cumulative Distribution Transform for Limited Data Recognition](https://mlsanity.com/paper/2512.08099/) (2025)

## Abstract (arXiv)

Data classification without access to labeled samples remains a challenging problem. It usually depends on an appropriately chosen distance between features, a topic addressed in metric learning. Recently, Huizing, Cantini and Peyr'e proposed to simultaneously learn optimal transport (OT) cost matrices between samples and features of the dataset. This leads to the task of finding positive eigenvectors of a certain nonlinear function that maps cost matrices to OT distances. Having this basic idea in mind, we consider both the algorithmic and the modeling part of unsupervised metric learning. First, we examine appropriate algorithms and their convergence. In particular, we propose to use the stochastic random function iteration algorithm and prove that it converges linearly for our setting, although our operators are not paracontractive as it was required for convergence so far. Second, we ask the natural question if the OT distance can be replaced by other distances. We show how Mahalanobis-like distances fit into our considerations. Further, we examine an approach via graph Laplacians. In contrast to the previous settings, we have just to deal with linear functions in the wanted matrices here, so that simple algorithms from linear algebra can be applied.