We study the multi-marginal partial optimal transport (POT) problem between $m$ discrete (unbalanced) measures with at most $n$ supports. We first prove that we can obtain two equivalence forms of the multimarginal POT problem in terms of the multimarginal optimal transport problem via novel extensions of cost tensor. The first equivalence form is derived under the assumptions that the total masses of each measure are sufficiently close while the second equivalence form does not require any conditions on these masses but at the price of more sophisticated extended cost tensor. Our proof techniques for obtaining these equivalence forms rely on novel procedures of moving mass in graph theory to push transportation plan into appropriate regions. Finally, based on the equivalence forms, we develop optimization algorithm, named ApproxMPOT algorithm, that builds upon the Sinkhorn algorithm for solving the entropic regularized multimarginal optimal transport. We demonstrate that the ApproxMPOT algorithm can approximate the optimal value of multimarginal POT problem with a computational complexity upper bound of the order $\tilde{\mathcal{O}}(m^3(n+1)^{m}/ \varepsilon^2)$ where $\varepsilon > 0$ stands for the desired tolerance.
翻译:我们首先通过成本高调的新扩展,从多边最佳运输问题的角度,证明我们可以通过多边最佳运输问题获得多边POT问题的两种等式形式。第一个等式形式来自以下假设:每项措施的总质量已经足够接近,而第二个等值形式并不要求这些质量的任何条件,而是要求以更复杂的延长成本为代价。我们获得这些等值形式的证明技术依赖于图表理论中的移动质量的新程序,将运输计划推入适当的区域。最后,根据等式形式,我们开发了优化算法,名为ApproxMPOT算法,该算法以Sinkhorn算法为基础,用于解决昆虫的正规化多边际最佳运输问题。我们证明,ApproxMPOT算法可以与多边基POT问题的最佳价值相近,其计算复杂性为 $\ tilde_mathcal {O_(m3+1)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\