Unsupervised Ground Metric Learning using Wasserstein Eigenvectors

• URL: http://arxiv.org/abs/2102.06278v1
• Date: Thu, 11 Feb 2021 21:32:59 GMT
• Title: Unsupervised Ground Metric Learning using Wasserstein Eigenvectors
• Authors: Geert-Jan Huizing, Laura Cantini, Gabriel Peyr\'e
• Abstract summary: Key bottleneck is design of a "ground" cost which should be adapted to the task under study. 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. We also introduce a scalable computational method using entropic regularization, which operates a principal component analysis dimensionality reduction.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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