On the Complexity of Approximating Multimarginal Optimal Transport
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We study the complexity of approximating the multimarginal optimal transport (OT) problem, a generalization of the classical optimal transport distance, considered here between m discrete probability distributions supported each on n support points. First, we show that the multimarginal OT problem is not a minimum-cost flow problem when m ≥ 3. This implies that many of the combinatorial algorithms developed for classical OT are not applicable to multimarginal OT, and therefore the standard interior-point algorithm bounds result in an intractable complexity bound of O(n^3m). Second, we propose and analyze two simple algorithms for approximating the multimarginal OT problem. The first algorithm, which we refer to as multimarginal Sinkhorn, improves upon previous multimarginal generalizations of the celebrated Sinkhorn algorithm. We show that it achieves a near-linear time complexity bound of O(m^3 n^m / ε^2) for a tolerance ε∈ (0, 1). This matches the best known complexity bound for the Sinkhorn algorithm when m = 2 for approximating the classical OT distance. The second algorithm, which we refer to as multimarginal Randkhorn, accelerates the first algorithm by incorporating a randomized estimate sequence and achieves a complexity bound of O(m^8/3 n^m+1/3/ε). This improves on the complexity bound of the first algorithm by 1/ε and matches the best known complexity bound for the Randkhorn algorithm when m=2 for approximating the classical OT distance.
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