Polynomial-time algorithms for Multimarginal Optimal Transport problems with structure

08/07/2020
by   Jason M. Altschuler, et al.
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Multimarginal Optimal Transport (MOT) has recently attracted significant interest due to its many applications. However, in most applications, the success of MOT is severely hindered by a lack of sub-exponential time algorithms. This paper develops a general theory about "structural properties" that make MOT tractable. We identify two such properties: decomposability of the cost into either (i) local interactions and simple global interactions; or (ii) low-rank interactions and sparse interactions. We also provide strong evidence that (iii) repulsive costs make MOT intractable by showing that several such problems of interest are NP-hard to solve–even approximately. These three structures are quite general, and collectively they encompass many (if not most) current MOT applications. We demonstrate our results on a variety of applications in machine learning, statistics, physics, and computational geometry.

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