[论文解读] Computational Optimal Transport

Gabriel Peyré, Marco Cuturi|arXiv (Cornell University)|Mar 1, 2018

Data Management and Algorithms被引用 40

一句话总结

对最优运输理论的全面综述，聚焦计算方法，包括 Kantorovich 松弛、熵正则化，以及离散和连续测度的可扩展算法。

ABSTRACT

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

研究动机与目标

• 解释最优运输的理论基础及其与概率空间几何的联系。
• 介绍求解 OT 问题的计算框架和算法，强调对大规模问题的可扩展性。
• 连接离散与连续的 OT，并讨论数据科学应用中的实际计算。
• 综述 OT 的泛化与扩展，并将其与相关统计与信息理论方法相连接。

提出的方法

• 引入直方图与测度，并在 Monge 与 Kantorovich 形式下定义 OT 问题。
• 推导 Kantorovich 松弛为关于耦合的线性规划，使质量分割成为可能。
• 讨论对偶形式、基于网络的和拍卖型算法，以及运输计划的结构。
• 描述熵正则化与 Sinkhorn 算法，包括在对数域中的稳定性实现。
• 涵盖半离散 OT、W1、动态形式，以及诸如 Gromov–Wasserstein 与切片运输等扩展。

实验结果

研究问题

• RQ1 如何将最优运输公式化并高效求解大规模离散与连续测度？
• RQ2 哪些关键算法策略（例如 Kantorovich 松弛、熵正则化）能够实现可扩展的 OT 计算？
• RQ3 离散与连续 OT 如何相关，以及它们在数据科学应用中的实际桥接？
• RQ4 OT 的重要扩展和泛化有哪些，以及它们如何与相关推断与信息理论概念相连接？

主要发现

• OT 可以通过 Monge 映射或 Kantorovich 耦合来形式化，后者使质量分割和凸优化成为可能。
• Kantorovich 的公式导致一个关于 Birkhoff 多面体的线性规划，确保计算可行性和解的存在性。
• 熵正则化带来更快、可扩展的算法（如 Sinkhorn），并在对数域中实现稳定。
• 该书/专著将 OT 与半离散场景、Wp 距离、动态形式以及如 Gromov–Wasserstein 和切片 OT 等扩展联系起来。
• 该工作强调 OT 理论与数值方法、核方法和信息理论在实际数据科学问题中的相互作用。