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2023•arXiv (Cornell University)Open access

Extending the Wasserstein metric to positive measures

Hugo Leblanc, Thibaut Le Gouic, Jacques Liandrat, Magali Tournus

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Abstract

We define a metric in the space of positive finite positive measures that extends the 2-Wasserstein metric, i.e. its restriction to the set of probability measures is the 2-Wasserstein metric. We prove a dual and a dynamic formulation and extend the gradient flow machinery of the Wasserstein space. In addition, we relate the barycenter in this space to the barycenter in the Wasserstein space of the normalized measures.

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We define a metric in the space of positive finite positive measures that extends the 2-Wasserstein metric, i.e. its restriction to the set of probability measures is the 2-Wasserstein metric. We prove a dual and a dynamic formulation and extend the gradient flow machinery of the Wasserstein space. In addition, we relate the barycenter in this space to the barycenter in the Wasserstein space of the normalized measures.

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Available abstract

We define a metric in the space of positive finite positive measures that extends the 2-Wasserstein metric, i.e. its restriction to the set of probability measures is the 2-Wasserstein metric. We prove a dual and a dynamic formulation and extend the gradient flow machinery of the Wasserstein space. In addition, we relate the barycenter in this space to the barycenter in the Wasserstein space of the normalized measures.

Key concepts: Wasserstein metric, Metric (unit), Mathematics, Probability measure, Metric space, Space (punctuation), Set (abstract data type), Stein's method

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