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Wasserstein distance to independence models. (English) Zbl 1460.62205
Summary: An independence model for discrete random variables is a Segre-Veronese variety in a probability simplex. Any metric on the set of joint states of the random variables induces a Wasserstein metric on the probability simplex. The unit ball of this polyhedral norm is dual to the Lipschitz polytope. Given any data distribution, we seek to minimize its Wasserstein distance to a fixed independence model. The solution to this optimization problem is a piecewise algebraic function of the data. We compute this function explicitly in small instances, we study its combinatorial structure and algebraic degrees in general, and we present some experimental case studies.
MSC:
62R01 Algebraic statistics
60C05 Combinatorial probability
62B05 Sufficient statistics and fields
Software:
Macaulay2; SCIP
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