标题： 最优传输的全概率设计 显示英文标题

标题： Fully Probabilistic Design for Optimal Transport

摘要： 本文的目标是引入最优传输（Optimal Transport, OT）的一个新理论框架，使用全概率设计（Fully Probabilistic Design, FPD）的术语和技术。 最优传输是一种比较概率测度的经典方法，并已在广泛领域成功应用（计算机视觉 Rubner 等人 [2004]，计算机图形学 Solomon 等人 [2015]，自然语言处理 Kusner 等人 [2015] 等）。 然而，我们认为当前的 OT 框架存在两个缺点：首先，在 OT 问题中难以诱导通用约束和概率知识；其次，当前的形式化没有解决边缘不确定性的问题，因此缺乏设计鲁棒解的机制。 通过将 OT 问题视为具有边缘约束的概率密度函数的最优设计，我们证明了 OT 是更通用的 FPD 框架的一个实例。 在这个新的设定下，我们可以为 OT 框架配备处理概率约束和推导不确定性量化器所需的机制，从而建立一个新的扩展框架，称为 FPD-OT。 本文的主要贡献在于建立了 OT 和 FPD 之间的联系，为两者提供了新的理论见解。 这将为后续工作中 FPD-OT 的应用奠定基础，特别是在处理更复杂的知识约束以及在边缘不确定情况下设计鲁棒解方面。 显示英文摘要

摘要： The goal of this paper is to introduce a new theoretical framework for Optimal Transport (OT), using the terminology and techniques of Fully Probabilistic Design (FPD). Optimal Transport is the canonical method for comparing probability measures and has been successfully applied in a wide range of areas (computer vision Rubner et al. [2004], computer graphics Solomon et al. [2015], natural language processing Kusner et al. [2015], etc.). However, we argue that the current OT framework suffers from two shortcomings: first, it is hard to induce generic constraints and probabilistic knowledge in the OT problem; second, the current formalism does not address the question of uncertainty in the marginals, lacking therefore the mechanisms to design robust solutions. By viewing the OT problem as the optimal design of a probability density function with marginal constraints, we prove that OT is an instance of the more generic FPD framework. In this new setting, we can furnish the OT framework with the necessary mechanisms for processing probabilistic constraints and deriving uncertainty quantifiers, hence establishing a new extended framework, called FPD-OT. Our main contribution in this paper is to establish the connection between OT and FPD, providing new theoretical insights for both. This will lay the foundations for the application of FPD-OT in a subsequent work, notably in processing more sophisticated knowledge constraints, as well as in designing robust solutions in the case of uncertain marginals.