Computing All Optimal Partial $p$-Wasserstein Matchings on the Line

2026-08-19

AI Summary

本文解决了计算一维离散分布间所有最优部分$p$-Wasserstein匹配的问题，提出了一种基于FFT的数据结构方法，将时间复杂度降低至$O(pn\log^2 n)$。

Abstract

For $p \ge 1$, the $p$-Wasserstein distance measures the minimum cost of transporting probability mass between distributions, where moving unit mass between two points costs the $p$th power of their distance. For discrete distributions in one dimension, full transport is especially simple: after sorting, mass is matched in order along the line. By contrast, partial and unbalanced transport on the line remains much less understood. Recently, Chapel and Tavenard [ICLR'25] showed that, for $p=1$, all optimal partial transport plans between distributions supported on $n$ points, with uniform mass at each point, can be computed in $O(n\log n)$ time by exploiting the metric structure of the cost. For $p>1$, this structure no longer applies, and existing approaches require $Ω(n^2)$ time. Our main contribution is an FFT-based data structure for balanced-interval transport queries, which bypasses this quadratic bottleneck and yields an $O(p\,n\log^2 n)$-time algorithm for computing all optimal partial transports on the line for every finite $p\ge 1$. We also provide an open-source C++ implementation that outperforms the state-of-the-art baseline on a range of synthetic instances. Finally, we establish a conditional lower bound for $p=\infty$: any subquadratic-time algorithm for computing all optimal partial transport plan costs on the line would violate the $(\min,+)$-Convolution Hypothesis. This separates the problem from full optimal transport, which is solvable in $O(n\log n)$.

Problem

• p-Wasserstein distance
• partial transport
• one-dimensional distributions
• computational complexity
• FFT-based data structure