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In this work, we provide faster algorithms for approximating the optimal transport distance, e.g. earth mover's distance, between two discrete probability distributions $\mu, \nu \in \Delta^n$. Given a cost function $C : [n] \times [n] \to…

Data Structures and Algorithms · Computer Science 2020-01-29 Jose Blanchet , Arun Jambulapati , Carson Kent , Aaron Sidford

We investigate the problem of efficiently computing optimal transport (OT) distances, which is equivalent to the node-capacitated minimum cost maximum flow problem in a bipartite graph. We compare runtimes in computing OT distances on data…

Data Structures and Algorithms · Computer Science 2020-07-07 Yihe Dong , Yu Gao , Richard Peng , Ilya Razenshteyn , Saurabh Sawlani

Transportation cost is an attractive similarity measure between probability distributions due to its many useful theoretical properties. However, solving optimal transport exactly can be prohibitively expensive. Therefore, there has been…

Machine Learning · Computer Science 2019-11-04 Nathaniel Lahn , Deepika Mulchandani , Sharath Raghvendra

Efficient computation of the optimal transport distance between two distributions serves as an algorithm subroutine that empowers various applications. This paper develops a scalable first-order optimization-based method that computes…

Machine Learning · Computer Science 2024-06-21 Gen Li , Yanxi Chen , Yu Huang , Yuejie Chi , H. Vincent Poor , Yuxin Chen

We analyze two algorithms for approximating the general optimal transport (OT) distance between two discrete distributions of size $n$, up to accuracy $\varepsilon$. For the first algorithm, which is based on the celebrated Sinkhorn's…

Data Structures and Algorithms · Computer Science 2018-06-08 Pavel Dvurechensky , Alexander Gasnikov , Alexey Kroshnin

The current best practice for computing optimal transport (OT) is via entropy regularization and Sinkhorn iterations. This algorithm runs in quadratic time as it requires the full pairwise cost matrix, which is prohibitively expensive for…

Machine Learning · Computer Science 2022-04-06 Johannes Gasteiger , Marten Lienen , Stephan Günnemann

Given a $d$-dimensional continuous (resp. discrete) probability distribution $\mu$ and a discrete distribution $\nu$, the semi-discrete (resp. discrete) Optimal Transport (OT) problem asks for computing a minimum-cost plan to transport mass…

Computational Geometry · Computer Science 2023-11-07 Pankaj K. Agarwal , Sharath Raghvendra , Pouyan Shirzadian , Keegan Yao

We develop a fast and reliable method for solving large-scale optimal transport (OT) problems at an unprecedented combination of speed and accuracy. Built on the celebrated Douglas-Rachford splitting technique, our method tackles the…

Optimization and Control · Mathematics 2021-10-25 Vien V. Mai , Jacob Lindbäck , Mikael Johansson

Recent advances in label assignment in object detection mainly seek to independently define positive/negative training samples for each ground-truth (gt) object. In this paper, we innovatively revisit the label assignment from a global…

Computer Vision and Pattern Recognition · Computer Science 2021-03-29 Zheng Ge , Songtao Liu , Zeming Li , Osamu Yoshie , Jian Sun

Entropic optimal transport (OT) and the Sinkhorn algorithm have made it practical for machine learning practitioners to perform the fundamental task of calculating transport distance between statistical distributions. In this work, we focus…

Optimization and Control · Mathematics 2024-03-11 Xun Tang , Holakou Rahmanian , Michael Shavlovsky , Kiran Koshy Thekumparampil , Tesi Xiao , Lexing Ying

Optimal Transport (OT) distances are now routinely used as loss functions in ML tasks. Yet, computing OT distances between arbitrary (i.e. not necessarily discrete) probability distributions remains an open problem. This paper introduces a…

Optimization and Control · Mathematics 2020-07-03 Arthur Mensch , Gabriel Peyré

In machine learning, Optimal Transport (OT) theory is extensively utilized to compare probability distributions across various applications, such as graph data represented by node distributions and image data represented by pixel…

Machine Learning · Computer Science 2025-01-30 Jianming Huang , Xun Su , Zhongxi Fang , Hiroyuki Kasai

Optimal transportation, or computing the Wasserstein or ``earth mover's'' distance between two distributions, is a fundamental primitive which arises in many learning and statistical settings. We give an algorithm which solves this problem…

Data Structures and Algorithms · Computer Science 2019-06-04 Arun Jambulapati , Aaron Sidford , Kevin Tian

Optimal transport (OT) is a powerful geometric and probabilistic tool for finding correspondences and measuring similarity between two distributions. Yet, its original formulation relies on the existence of a cost function between the…

Machine Learning · Statistics 2020-11-09 Ievgen Redko , Titouan Vayer , Rémi Flamary , Nicolas Courty

Computing optimal transport (OT) distances such as the earth mover's distance is a fundamental problem in machine learning, statistics, and computer vision. In this paper, we study the problem of approximating the general OT distance…

Data Structures and Algorithms · Computer Science 2023-01-18 Zhao Song , Tianyi Zhou

We study the complexity of approximating the multimarginal optimal transport (MOT) distance, a generalization of the classical optimal transport distance, considered here between $m$ discrete probability distributions supported each on $n$…

Machine Learning · Statistics 2022-02-23 Tianyi Lin , Nhat Ho , Marco Cuturi , Michael I. Jordan

We provide a computational complexity analysis for the Sinkhorn algorithm that solves the entropic regularized Unbalanced Optimal Transport (UOT) problem between two measures of possibly different masses with at most $n$ components. We show…

Computational Complexity · Computer Science 2020-11-20 Khiem Pham , Khang Le , Nhat Ho , Tung Pham , Hung Bui

We propose a simple subsampling scheme for fast randomized approximate computation of optimal transport distances. This scheme operates on a random subset of the full data and can use any exact algorithm as a black-box back-end, including…

Computation · Statistics 2020-12-17 Max Sommerfeld , Jörn Schrieber , Yoav Zemel , Axel Munk

Optimal transport (OT) defines a powerful framework to compare probability distributions in a geometrically faithful way. However, the practical impact of OT is still limited because of its computational burden. We propose a new class of…

Optimization and Control · Mathematics 2016-05-30 Genevay Aude , Marco Cuturi , Gabriel Peyré , Francis Bach

Optimal transport (OT) aims to find a map $T$ that transports mass from one probability measure to another while minimizing a cost function. Recently, neural OT solvers have gained popularity in high dimensional biological applications such…

Machine Learning · Computer Science 2025-05-20 Peter Chen , Yue Xie , Qingpeng Zhang
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