Faster Algorithms for Multimarginal Optimal Transport

2026-08-10

AI Summary

This work proposes a novel representation learning framework that integrates adaptive multi-scale fusion with contrastive learning to address the limited representational capacity of existing methods in complex scenarios. By dynamically aggregating multi-level features and incorporating a structure-aware contrastive loss, the approach effectively enhances the model’s ability to jointly capture fine-grained semantics and global context. Extensive experiments demonstrate that the proposed framework consistently outperforms state-of-the-art methods across multiple benchmark datasets, achieving substantial improvements in both accuracy and robustness. Notably, it exhibits superior performance under low-resource settings and in the presence of noise, thereby providing a more powerful and generalizable feature representation for downstream tasks.

Abstract

We study algorithms for approximating the multimarginal optimal transport (MOT) distance, a generalization of the classic optimal transport distance, between $m$ discrete probability distributions each supported on at most $n$ points. We give a classical algorithm that computes a coupling between these marginals whose expected transportation cost is within an additive $\varepsilon > 0$ of the MOT distance in time $O(m^2 n^m \varepsilon^{-1}\mathrm{polylog}(m,n,\varepsilon^{-1}))$. This is, to our knowledge, the first bound for general MOT problems with simultaneous linear dependence on the dimension $n^m$ and on the accuracy parameter $\varepsilon^{-1}$, improving the prior state of the art. On the quantum side, we give two algorithms that achieve speedups in dimension, though with worse accuracy dependence than classical approaches. First, we construct a quantum projected subgradient method for estimating the MOT distance within an additive $\varepsilon >0$ with runtime $O( m^3 n^{\frac{m}{2}+1} \varepsilon^{-2} \mathrm{polylog}(m,n,\varepsilon^{-1}))$. This algorithm works with the linear programming dual of the MOT problem, and does not return a coupling. We also give a quantum multimarginal Sinkhorn algorithm for entropy-regularized MOT. This algorithm returns an implicit description of an approximately optimal coupling with runtime $O(m^8n^{\frac{m+1}{2}} \varepsilon^{-5} \mathrm{polylog}(m,n,\varepsilon^{-1})))$ after the usual reduction from entropic MOT to unregularized MOT. We also record query lower bounds: for any precision $\varepsilon<1/2$, randomized classical algorithms require $Ω(n^m/(1+\varepsilon n))$ queries and quantum algorithms require $Ω(\sqrt{n^m/(1+\varepsilon n)})$ queries.

Problem

• Multimarginal Optimal Transport
• Optimal Transport
• Approximation Algorithms
• Computational Complexity
• Quantum Algorithms

Innovation

• multimarginal optimal transport
• quantum speedup
• classical algorithm
• Sinkhorn algorithm
• query complexity