# Global Ground Metric Learning with Applications to scRNA data

arXiv: https://arxiv.org/abs/2506.15383 · HTML: https://mlsanity.com/paper/2506.15383/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 57 references that are in the map.
- First posted on arXiv 18 Jun 2025.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Riemannian Metric Learning via Optimal Transport](https://mlsanity.com/paper/2205.09244/) (2022)
- [metric-learn: Metric Learning Algorithms in Python](https://mlsanity.com/paper/1908.04710/) (2019)
- [Re-evaluating Word Mover's Distance](https://mlsanity.com/paper/2105.14403/) (2021)
- [Learning Local Metrics and Influential Regions for Classification](https://mlsanity.com/paper/1802.03452/) (2018)
- [Unsupervised Ground Metric Learning using Wasserstein Singular Vectors](https://mlsanity.com/paper/2102.06278/) (2021)
- [Lipschitz Continuity of Mahalanobis Distances and Bilinear Forms](https://mlsanity.com/paper/1604.01376/) (2016)

## Abstract (arXiv)

Optimal transport provides a robust framework for comparing probability distributions. Its effectiveness is significantly influenced by the choice of the underlying ground metric. Traditionally, the ground metric has either been (i) predefined, e.g., as the Euclidean distance, or (ii) learned in a supervised way, by utilizing labeled data to learn a suitable ground metric for enhanced task-specific performance. Yet, predefined metrics typically cannot account for the inherent structure and varying importance of different features in the data, and existing supervised approaches to ground metric learning often do not generalize across multiple classes or are restricted to distributions with shared supports. To address these limitations, we propose a novel approach for learning metrics for arbitrary distributions over a shared metric space. Our method provides a distance between individual points like a global metric, but requires only class labels on a distribution-level for training. The learned global ground metric enables more accurate optimal transport distances, leading to improved performance in embedding, clustering and classification tasks. We demonstrate the effectiveness and interpretability of our approach using patient-level scRNA-seq data spanning multiple diseases.