Polynomial-time algorithms for multimarginal optimal transport problems with structure

Multimarginal Optimal Transport (MOT) has 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, MOT in general requires exponential time in...

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Veröffentlicht in:Mathematical programming Jg. 199; H. 1-2; S. 1107 - 1178
Hauptverfasser: Altschuler, Jason M., Boix-Adserà, Enric
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2023
Springer
Schlagworte:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Machine learning
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
Implicit linear programming
90C06
Polynomial-time algorithms
Structured linear programs
Multimarginal optimal transport
90C08
ISSN:0025-5610, 1436-4646
Online-Zugang:Volltext
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Zusammenfassung:Multimarginal Optimal Transport (MOT) has 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, MOT in general requires exponential time in the number of marginals k and their support sizes n . This paper develops a general theory about what “structure” makes MOT solvable in poly ( n , k ) time. We develop a unified algorithmic framework for solving MOT in poly ( n , k ) time by characterizing the structure that different algorithms require in terms of simple variants of the dual feasibility oracle. This framework has several benefits. First, it enables us to show that the Sinkhorn algorithm, which is currently the most popular MOT algorithm, requires strictly more structure than other algorithms do to solve MOT in poly ( n , k ) time. Second, our framework makes it much simpler to develop poly ( n , k ) time algorithms for a given MOT problem. In particular, it is necessary and sufficient to (approximately) solve the dual feasibility oracle—which is much more amenable to standard algorithmic techniques. We illustrate this ease-of-use by developing poly ( n , k ) -time algorithms for three general classes of MOT cost structures: (1) graphical structure; (2) set-optimization structure; and (3) low-rank plus sparse structure. For structure (1), we recover the known result that Sinkhorn has poly ( n , k ) runtime; moreover, we provide the first poly ( n , k ) time algorithms for computing solutions that are exact and sparse. For structures (2)-(3), we give the first poly ( n , k ) time algorithms, even for approximate computation. Together, these three structures encompass many—if not most—current applications of MOT.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-022-01868-7