[论文解读] Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form

Hicham Janati, Boris Muzellec+2 more|arXiv (Cornell University)|Jun 3, 2020

Probabilistic and Robust Engineering Design参考文献 46被引用 31

一句话总结

本文给出高斯测度之间熵正则化最优传输的封闭形式公式，包括非平衡情形，并表明最优传输计划是高斯分布，且 Sinkhorn 迭代的固定点表征可解。

ABSTRACT

Although optimal transport (OT) problems admit closed form solutions in a very few notable cases, e.g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry. On the other hand, the numerical resolution of OT problems using entropic regularization has given rise to many applications, but because there are no known closed-form solutions for entropic regularized OT problems, these approaches are mostly algorithmic, not informed by elegant closed forms. In this paper, we propose to fill the void at the intersection between these two schools of thought in OT by proving that the entropy-regularized optimal transport problem between two Gaussian measures admits a closed form. Contrary to the unregularized case, for which the explicit form is given by the Wasserstein-Bures distance, the closed form we obtain is differentiable everywhere, even for Gaussians with degenerate covariance matrices. We obtain this closed form solution by solving the fixed-point equation behind Sinkhorn's algorithm, the default method for computing entropic regularized OT. Remarkably, this approach extends to the generalized unbalanced case -- where Gaussian measures are scaled by positive constants. This extension leads to a closed form expression for unbalanced Gaussians as well, and highlights the mass transportation / destruction trade-off seen in unbalanced optimal transport. Moreover, in both settings, we show that the optimal transportation plans are (scaled) Gaussians and provide analytical formulas of their parameters. These formulas constitute the first non-trivial closed forms for entropy-regularized optimal transport, thus providing a ground truth for the analysis of entropic OT and Sinkhorn's algorithm.

研究动机与目标

• 将 entropic OT 作为具有非平衡传输能力的可扩展正则化进行动机阐述。
• 在熵正则化下推导平衡高斯 OT 的封闭式公式。
• 扩展到非平衡高斯 OT 并刻画质量传输/毁灭的权衡。
• 证明最优传输计划是高斯分布并给出显式参数化。
• 提供对封闭形式表达式的理论与实证验证。

提出的方法

• 使用 Sinkhorn 框架将带熵正则化与非平衡质量（KL 项）之 OT 进行形式化。
• 通过分析高斯族中的 Sinkhorn 迭代固定点（中心化、二次势）来获得封闭式表达。
• 通过矩阵（A、B，以及辅助的 F、G）推导 Sinkhorn 势和传输计划参数的显式表达。
• 证明最优计划在 R^d x R^d 上是高斯分布，并给出其均值/协方差，表示为 A、B 和正则化参数的函数。
• 将平衡结果扩展到非平衡情形，推导出 UOT_sigma 的封闭式解，包括总传输质量 m_pi。
• 提供辅助引理（Lemmas 1-2）和命题（Propositions 1-7）以支持固定点和基于行列式的计算。

实验结果

研究问题

• RQ1 在平衡和非平衡质量约束下，高斯测度之间的熵正则化 OT 是否可以用封闭式表达？
• RQ2 在熵正则化下，高斯之间的最优传输计划是否为高斯分布，其参数是否可以解析地计算？
• RQ3 非平衡质量约束如何改变高斯间 OT 的传输与毁灭之间的权衡？
• RQ4 Sinkhorn 迭代是否存在固定点形式，从而为高斯提供显式的封闭式势？
• RQ5 在高斯情形下，熵正则化 OT 成本的可微性和凸性性质是什么？

主要发现

• 得到高斯测度之间的 OT_sigma 的封闭式表达，显示距离等于均值平方距离与一个与高斯相关的项 B_sigma^2(A,B) 的和。
• 熵 OT 计划为高斯分布，具有由包含 A、B 与一个交叉项 C_sigma 的分块协方差矩阵给出的显式均值和协方差。
• 即使协方差矩阵为奇异，解也仍然良定义且可微分，这与 Bures 距离不同。
• 对于非平衡高斯 OT 推导出类似的封闭形式，产生具有质量 m_pi 的高斯传输计划和对偶势结构。
• 高斯 Sinkhorn 因子的一致方程简化为易处理的矩阵方程，能够直接计算最优势 (U, V) 和传输计划。
• 数值实验验证收敛到封闭式公式，并说明正则化和非平衡质量对传输计划的影响。