Unsupervised Ground Metric Learning using Wasserstein Singular Vectors
This addresses the fundamental problem of defining meaningful distances in datasets for data-driven applications of Optimal Transport, particularly in unsupervised settings where labeled data is unavailable.
The paper tackles the problem of unsupervised ground metric learning for Optimal Transport by proposing Wasserstein Singular Vectors, which simultaneously compute distances between samples and features without labeled data. It demonstrates this method on a single-cell RNA-sequencing dataset, providing scalable computational approximations.
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.