Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 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.

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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: physics