44 research outputs found
Batch Greenkhorn Algorithm for Entropic-Regularized Multimarginal Optimal Transport: Linear Rate of Convergence and Iteration Complexity
In this work we propose a batch multimarginal version of the Greenkhorn algorithm for the entropic-regularized optimal transport problem. This framework is general enough to cover, as particular cases, existing Sinkhorn and Greenkhorn algorithms for the bi-marginal setting, and greedy MultiSinkhorn for the general multimarginal case. We provide a comprehensive convergence analysis based on the properties of the iterative Bregman projections method with greedy control. Linear rate of convergence as well as explicit bounds on the iteration complexity are obtained. When specialized to the above mentioned algorithms, our results give new convergence rates or provide key improvements over the state-of-the-art rates. We present numerical experiments showing that the flexibility of the batch can be exploited to improve performance of Sinkhorn algorithm both in bi-marginal and multimarginal settings
Distributed optimization with quantization for computing Wasserstein barycenters
We study the problem of the decentralized computation of entropy-regularized semi-discrete Wasserstein barycenters over a network. Building upon recent primal-dual approaches, we propose a sampling gradient quantization scheme that allows efficient communication and computation of approximate barycenters where the factor distributions are stored distributedly on arbitrary networks. The communication and algorithmic complexity of the proposed algorithm are shown, with explicit dependency on the size of the support, the number of distributions, and the desired accuracy. Numerical results validate our algorithmic analysis
Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm
We study the fixed-support Wasserstein barycenter problem (FS-WBP), which
consists in computing the Wasserstein barycenter of discrete probability
measures supported on a finite metric space of size . We show first that the
constraint matrix arising from the standard linear programming (LP)
representation of the FS-WBP is \textit{not totally unimodular} when
and . This result resolves an open question pertaining to the
relationship between the FS-WBP and the minimum-cost flow (MCF) problem since
it proves that the FS-WBP in the standard LP form is not an MCF problem when and . We also develop a provably fast \textit{deterministic}
variant of the celebrated iterative Bregman projection (IBP) algorithm, named
\textsc{FastIBP}, with a complexity bound of
, where is the
desired tolerance. This complexity bound is better than the best known
complexity bound of for the IBP algorithm in
terms of , and that of from
accelerated alternating minimization algorithm or accelerated primal-dual
adaptive gradient algorithm in terms of . Finally, we conduct extensive
experiments with both synthetic data and real images and demonstrate the
favorable performance of the \textsc{FastIBP} algorithm in practice.Comment: Accepted by NeurIPS 2020; fix some confusing parts in the proof and
improve the empirical evaluatio
Sampling-Based Approaches for Multimarginal Optimal Transport Problems with Coulomb Cost
The multimarginal optimal transport problem with Coulomb cost arises in
quantum physics and is vital in understanding strongly correlated quantum
systems. Its intrinsic curse of dimensionality can be overcome with a
Monge-like ansatz. A nonconvex quadratic programmming then emerges after
employing discretization and penalty. To globally solve this nonconvex
problem, we adopt a grid refinements-based framework, in which a local solver
is heavily invoked and hence significantly determines the overall efficiency.
The block structure of this nonconvex problem suggests taking block coordinate
descent-type methods as the local solvers, while the existing ones can get
seriously afflicted with the poor scalability induced by the associated
sparse-dense matrix multiplications. In this work, borrowing the tools from
optimal transport, we develop novel methods that favor highly scalable schemes
for subproblems and are completely free of the full matrix multiplications
after introducing entrywise sampling. Convergence and asymptotic properties are
built on the theory of random matrices. The numerical results on several
typical physical systems corroborate the effectiveness and better scalability
of our approach, which also allows the first visualization for the approximate
optimal transport maps between electrons in three-dimensional contexts.Comment: 31 pages, 6 figures, 3 table
Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs
In this paper, we address the numerical solution to the multimarginal optimal
transport (MMOT) with pairwise costs. MMOT, as a natural extension from the
classical two-marginal optimal transport, has many important applications
including image processing, density functional theory and machine learning, but
yet lacks efficient and exact numerical methods. The popular
entropy-regularized method may suffer numerical instability and blurring
issues. Inspired by the back-and-forth method introduced by Jacobs and
L\'{e}ger, we investigate MMOT problems with pairwise costs. First, such
problems have a graphical representation and we prove equivalent MMOT problems
that have a tree representation. Second, we introduce a noval algorithm to
solve MMOT on a rooted tree, by gradient based method on the dual formulation.
Last, we obtain accurate solutions which can be used for the
regularization-free applications.Comment: Update the proof to Theorem 3.
Multi-marginal optimal transport: theory and applications
Over the past five years, multi-marginal optimal transport, a generalization
of the well known optimal transport problem of Monge and Kantorovich, has begun
to attract considerable attention, due in part to a wide variety of emerging
applications. Here, we survey this problem, addressing fundamental theoretical
questions including the uniqueness and structure of solutions. The (partial)
answers to these questions uncover a surprising divergence from the classical
two marginal setting, and reflect a delicate dependence on the cost function.
We go one to describe two applications of the multi-marginal problem.Comment: Typos corrected and minor changes to presentatio