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.
Çelik, T.Ö., Jamneshan, A., Montúfar, G., Sturmfels, B., Venturello, L. (2020). Optimal Transport to a Variety. In Mathematical Aspects of Computer and Information Sciences. MACIS 2019. (pp.364-381). Cham : Springer [10.1007/978-3-030-43120-4_29].
Optimal Transport to a Variety
Venturello, Lorenzo
2020-01-01
Abstract
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.
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https://hdl.handle.net/11365/1256097