Optimal Transport (OT) is a mathematical framework that first emerged in the eighteenth century and has led to a plethora of methods for answering many theoretical and applied questions. The last decade is a witness of the remarkable contributions of this classical optimization problem to machine learning. This paper is about where and how optimal transport is used in machine learning with a focus on the question of salable optimal transport. We provide a comprehensive survey of optimal transport while ensuring an accessible presentation as permitted by the nature of the topic and the context. First, we explain optimal transport background and introduce different flavors (i.e. mathematical formulations), properties, and notable applications. We then address the fundamental question of how to scale optimal transport to cope with the current demands of big and high dimensional data. We conduct a systematic analysis of the methods used in the literature for scaling OT and present the findings in a unified taxonomy. We conclude with presenting some open challenges and discussing potential future research directions. A live repository of related OT research papers is maintained in https://github.com/abdelwahed/OT_for_big_data.git.

翻译：最优传输（Optimal Transport, OT）是一种源于十八世纪的数学框架，已催生出众多解决理论与应用问题的方法。过去十年间，这一经典优化问题对机器学习领域做出了显著贡献。本文旨在探讨最优传输在机器学习中的运用场景与方式，重点关注其可扩展性问题。我们提供了一份关于最优传输的全面综述，同时力求在主题与语境的自然限制下，保持内容呈现的通俗易懂。首先，我们阐述了最优传输的背景，并介绍了其不同形式（即数学公式）、性质及重要应用。随后，我们聚焦于如何扩展最优传输以应对当前大数据与高维数据需求这一根本性问题。我们对文献中用于扩展最优传输的方法进行了系统分析，并将研究结果归纳为统一的分类体系。最后，我们提出了若干开放挑战，并探讨了潜在的未来研究方向。相关OT研究论文的实时资源库维护于 https://github.com/abdelwahed/OT_for_big_data.git。