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However, it is often desirable that the marginals of the optimal transport plan do not match the given measures exactly, which led to the introduction of the so-called unbalanced optimal transport. Since unbalanced methods were not examined for the multi-marginal setting so far, we address this topic in the present paper. More precisely, we introduce the unbalanced multi-marginal optimal transport problem and its dual and show that a unique optimal transport plan exists under mild assumptions. Furthermore, we generalize the Sinkhorn algorithm for regularized unbalanced optimal transport to the multi-marginal setting and prove its convergence. For cost functions decoupling according to a tree, the iterates can be computed efficiently. 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