  Optimal Transport (OT) defines geometrically meaningful "Wasserstein"
distances, used in machine learning applications to compare probability
distributions. However, a key bottleneck is the design of a "ground" cost which
should be adapted to the task under study. In most cases, supervised metric
learning is not accessible, and one usually resorts to some ad-hoc approach.
Unsupervised metric learning is thus a fundamental problem to enable
data-driven applications of Optimal Transport. In this paper, we propose for
the first time a canonical answer by computing the ground cost as a positive
eigenvector of the function mapping a cost to the pairwise OT distances between
the inputs. This map is homogeneous and monotone, thus framing unsupervised
metric learning as a non-linear Perron-Frobenius problem. We provide criteria
to ensure the existence and uniqueness of this eigenvector. In addition, we
introduce a scalable computational method using entropic regularization, which
- in the large regularization limit - operates a principal component analysis
dimensionality reduction. We showcase this method on synthetic examples and
datasets. Finally, we apply it in the context of biology to the analysis of a
high-throughput single-cell RNA sequencing (scRNAseq) dataset, to improve cell
clustering and infer the relationships between genes in an unsupervised way.
