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 problemsof 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 .
Multimarginal Optimal Transport (MOT) has recently attracted significant interest due to applications in machine learning, statistics, and the sciences . However, in most applications, the success of MOT is severely limited by a lack of efficient algorithms. Indeed, in general, MOT requires exponential time in the number of marginals k and their support sizes n. This paper develops a general theory about "structural properties" that make MOT solvable in poly(n,k) time . We identify two such properties: decomposability of the cost into either (i) local and simple global interactions; or (ii) low-rank and sparse components. These two structures encompass many--if not most--current applications of MOT. In addition to providing the first poly(n,k)-time algorithms for a wide range of MOT problems, our results also provide better algorithms for MOT problems that are already known to be tractable: Our algorithms compute solutions which are exact and sparse. (Previous algorithms can do neither.) We demonstrate our results theoretically and numerically on popular applications in machine learning, statistics, and fluid dynamics .
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its many applications
applications in machine learning, statistics, and the sciences
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hindered
limited
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sub-exponential time algorithms.
efficient algorithms. Indeed, in general, MOT requires exponential time in the number of marginals k and their support sizes n.
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tractable
solvable in poly(n,k) time
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interactions
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interactions and sparse interactions. We also provide strong evidence that (iii)repulsive costs make MOT intractable by showing that several such problemsof 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.
and sparse components. These two structures encompass many--if not most--current applications of MOT. In addition to providing the first poly(n,k)-time algorithms for a wide range of MOT problems, our results also provide better algorithms for MOT problems that are already known to be tractable: Our algorithms compute solutions which are exact and sparse. (Previous algorithms can do neither.)
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on a variety of
theoretically and numerically on popular
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physics, and computational geometry
and fluid dynamics