Optimal Transport to a Variety
Türkü Özlüm Çelik, Asgar Jamneshan, Guido Montúfar, Bernd Sturmfels, and Lorenzo Venturello
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Submission date: 09. Mar. 2021
Keywords and phrases: algebraic statistics, Linear Programming, Optimal Transport Estimator, Polynomial Optimization, Transportation Polytope, Triangulation, Wasserstein distance
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Link to arXiv: See the arXiv entry of this preprint.
DOI number (of the published article): 10.1007/978-3-030-43120-4_29
We study the problem of minimizing the Wasserstein distance between a probability distribution and an algebraic variety. We consider the setting of finite state spaces and describe the solution depending on the choice of the ground metric and the given distribution. The Wasserstein distance between the distribution and the variety is the minimum of a linear functional over a union of transportation polytopes. We obtain a description in terms of the solutions of a finite number of systems of polynomial equations. The case analysis is based on the ground metric. A detailed analysis is given for the two bit independence model.