论文标题
质量最佳运输
Constrained Mass Optimal Transport
论文作者
论文摘要
最佳的质量运输，也称为地球搬运工的问题，是各种学科中重要应用的优化问题，包括经济学，概率理论，流体动力学，宇宙学和地球物理学。最佳运输还发现了在图像注册，基于内容的图像检索中的成功应用程序，并且更普遍地在模式识别和机器学习中，作为测量数据之间差异的一种方式。本文介绍了约束最佳运输的问题。更准确地说，流体动力学方法被用作开发点，通过将软性约束对密度和动量场施加软性约束或将其限制为满足某些规定条件的曲线的子集来定义约束问题的起点。引入了一种算法系列来解决一类约束的鞍点问题，该问题在欧几里得空间的封闭凸子集上凸出了最佳的最佳运输，作为一种特殊情况。提出了收敛证明和数值结果。
Optimal mass transport, also known as the earth mover's problem, is an optimization problem with important applications in various disciplines, including economics, probability theory, fluid dynamics, cosmology and geophysics to cite a few. Optimal transport has also found successful applications in image registration, content-based image retrieval, and more generally in pattern recognition and machine learning as a way to measure dissimilarity among data. This paper introduces the problem of constrained optimal transport. The time-dependent formulation, more precisely, the fluid dynamics approach is used as a starting point from which the constrained problem is defined by imposing a soft constraint on the density and momentum fields or restricting them to a subset of curves that satisfy some prescribed conditions. A family of algorithms is introduced to solve a class of constrained saddle point problems, which has convexly constrained optimal transport on closed convex subsets of the Euclidean space as a special case. Convergence proofs and numerical results are presented.
