# Unsupervised Ground Metric Learning using Wasserstein Singular Vectors

arXiv: https://arxiv.org/abs/2102.06278 · HTML: https://mlsanity.com/paper/2102.06278/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 77 references that are in the map.
- First posted on arXiv 11 Feb 2021.

## Builds on (most cited first, 26 of 26)

- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [A Survey on Metric Learning for Feature Vectors and Structured Data](https://mlsanity.com/paper/1306.6709/) (2013)
- [Hierarchical Optimal Transport for Document Representation](https://mlsanity.com/paper/1906.10827/) (2019)
- [Fast Optimal Transport Averaging of Neuroimaging Data](https://mlsanity.com/paper/1503.08596/) (2015)
- [Inverse optimal transport](https://mlsanity.com/paper/1905.03950/) (2019)
- [Wasserstein Discriminant Analysis](https://mlsanity.com/paper/1608.08063/) (2016)
- [Learning to Match via Inverse Optimal Transport](https://mlsanity.com/paper/1802.03644/) (2018)
- [Spatio-Temporal Alignments: Optimal transport through space and time](https://mlsanity.com/paper/1910.03860/) (2019)
- [Ground Metric Learning on Graphs](https://mlsanity.com/paper/1911.03117/) (2019)
- [Regularized Optimal Transport is Ground Cost Adversarial](https://mlsanity.com/paper/2002.03967/) (2020)
- [Estimating matching affinity matrix under low-rank constraints](https://mlsanity.com/paper/1612.09585/) (2016)
- [The Gene Mover's Distance: Single-cell similarity via Optimal Transport](https://mlsanity.com/paper/2102.01218/) (2021)
- [SISTA: learning optimal transport costs under sparsity constraints](https://mlsanity.com/paper/2009.08564/) (2020)
- [Geometry of Graph Partitions via Optimal Transport](https://mlsanity.com/paper/1910.09618/) (2019)
- [A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors](https://mlsanity.com/paper/1812.09871/) (2018)

## Built on by (most cited first, 5 of 5)

- [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)
- [Fast unsupervised ground metric learning with tree-Wasserstein distance](https://mlsanity.com/paper/2411.07432/) (2024)
- [Unsupervised Ground Metric Learning](https://mlsanity.com/paper/2507.13094/) (2025)
- [Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization](https://mlsanity.com/paper/2410.02628/) (2024)
- [Global Ground Metric Learning with Applications to scRNA data](https://mlsanity.com/paper/2506.15383/) (2025)

## Abstract (arXiv)

Defining meaningful distances between samples in a dataset is a fundamental problem in machine learning. Optimal Transport (OT) lifts a distance between features (the "ground metric") to a geometrically meaningful distance between samples. However, there is usually no straightforward choice of ground metric. Supervised ground metric learning approaches exist but require labeled data. In absence of labels, only ad-hoc ground metrics remain. Unsupervised ground metric learning is thus a fundamental problem to enable data-driven applications of OT. In this paper, we propose for the first time a canonical answer by simultaneously computing an OT distance between samples and between features of a dataset. These distance matrices emerge naturally as positive singular vectors of the function mapping ground metrics to OT distances. We provide criteria to ensure the existence and uniqueness of these singular vectors. We then introduce scalable computational methods to approximate them in high-dimensional settings, using stochastic approximation and entropic regularization. Finally, we showcase Wasserstein Singular Vectors on a single-cell RNA-sequencing dataset.