多边际最优运输的更快算法

中文摘要 AI 辅助

我们研究多边际最优运输（Multimarginal Optimal Transport, MOT）距离的近似算法，MOT是经典最优运输距离的推广，针对m个离散概率分布，每个分布最多支撑在n个点上。我们提出一种经典算法，该算法计算这些边际之间的耦合，其期望运输成本与MOT距离的加性误差ε>0在时间O(m²nᵐε⁻¹polylog(m,n,ε⁻¹))内。据我们所知，这是首个针对一般MOT问题同时线性依赖于维度nᵐ和精度参数ε⁻¹的复杂度界，优于现有技术水平。在量子方面，我们提出两种在维度上实现加速但精度依赖弱于经典方法的算法：第一种是量子投影次梯度法，用于估计加性误差ε>0内的MOT距离，运行时间为O(m³n^(m/2+1)ε⁻²polylog(m,n,ε⁻¹))，该算法基于MOT问题的线性规划对偶，不返回耦合；第二种是熵正则化MOT的量子多边际Sinkhorn算法，在将熵正则化MOT归约为非正则化MOT后，返回近似最优耦合的隐式描述，运行时间为O(m⁸n^((m+1)/2)ε⁻⁵polylog(m,n,ε⁻¹))。我们还给出查询下界：对于任意精度ε<1/2，随机经典算法需要Ω(nᵐ/(1+εn))次查询，量子算法需要Ω(√(nᵐ/(1+εn)))次查询。

英文摘要

We study constructive discrete multimarginal optimal transport (MOT) among $m$ distributions on $n$ points, where the cost tensor $C$ has $N=n^m$ entries. For additive accuracy $\varepsilon$, let $κ=\max\{1,(\max C-\min C)/\varepsilon\}$. Classically, we give two algorithms that return exactly feasible additive-$\varepsilon$ couplings. A deterministic box--simplex method with rounding runs in $O(m^2Nκ\log N)=\widetilde O(m^3Nκ)$ time, while an exact reduction to positive packing followed by rank-one completion runs in randomized time $\widetilde O(m^2Nκ)$. At fixed $κ$, both match the $Ω(N)$ cost of writing a dense coupling, up to factors in $m$ and logarithms. Quantumly, in the general entry-access model, tensor Sinkhorn followed by sparse recovery returns an exactly feasible additive-$\varepsilon$ coupling as a classical list of $\widetilde O(m^2nκ^2)$ atoms, using $\widetilde O(m^4\sqrt{Nn}\,κ^3)$ coherent cost queries without materializing the tensor. Finally, for fixed $m$ and constant normalized accuracy, we prove sparse-output lower bounds of $\widetildeΩ(N)$ randomized classical queries, even with unrestricted output size, and $\widetildeΩ(\sqrt{Nn})$ quantum queries for outputs with at most $n\operatorname{polylog}(n)$ atoms. Hence, for fixed $m$ and normalized accuracy between $(\log n)^{-O(1)}$ and a sufficiently small constant, explicit sparse MOT construction has query complexity $\widetildeΘ(N)$ classically and $\widetildeΘ(\sqrt{Nn})$ quantumly.

发表机构

• Global Technology Applied Research, JPMorganChase（摩根大通全球科技应用研究）
• Quantitative Trading & Research, JPMorganChase（摩根大通量化交易与研究）

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