Matricial Wasserstein-1 Distance

2017/02/25 by Yongxin Chen, Tryphon T. Georgiou, Chen, Yongxin +5 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math-ph #math.FA #math.MP

paper · pdf · doi:10.48550/arxiv.1702.07921

8 pages

arxiv created 2017/03/04 · arxiv updated 2017/03/07

Abstract

In this note, we propose an extension of the Wasserstein 1-metric (W₁) for matrix probability densities, matrix-valued density measures, and an unbalanced interpretation of mass transport. The key is using duality theory, in particular, a "dual of the dual" formulation of W₁. This matrix analogue of the Earth Mover's Distance has several attractive features including ease of computation.

Citations

• A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
• Ricci curvature of Markov chains on metric spaces

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• Metrics on completely positive maps via noncommutative geometry

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