Optimal Transport to a Variety
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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.
KeywordsAlgebraic statistics Linear programming Optimal transport estimator Polynomial optimization Transportation Polytope Triangulation Wasserstein distance
GM has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no 757983).
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