1. 概念(蒙日问题): 蒙日(Monge)两百多前提出,将一个沙盘的沙子以最小花销传输到另外一个沙盘。另一种直观理解:把n个仓库的原料以最小代价传输到m个工厂。 2. Kantorovich Relaxation(Kantorovich问题):…
Optimal Transport (OT) 是一种数学理论,用于描述在两个不同分布之间进行数据传输的最优方式。这种理论可以用于多种应用场景,如图像处理、机器学习、统计学和计算流体力学等。 在OT理论中,我们把两个分布看作为两个“质量”分布。我们要找到一种传输方式,使得在保持两个分布的“质量”不变的前提下,能够最有效地在...
Optimal Transport (OT) 是一种数学理论,用于描述在两个不同分布之间进行数据传输的最优方式。这种理论可以用于多种应用场景,如图像处理、机器学习、统计学和计算流体力学等。 在OT理论中,我们把两个分布看作为两个“质量”分布。我们要找到一种传输方式,使得在保持两个分布的“质量”不变的前提下,能够最有效地在...
在例子中，对于离散概率向量，耦合矩阵P描述了传输过程。Wasserstein距离则基于Kantorovich最优传输问题，对n=m时，给出了一种不全面理解。它定义了一个距离，即对给定的点对距离（成本）求解一个类Kantorovich OT问题，使得两个分布以概率P加权的距离求和最小。熵（Entropic）正则化是处理大规模问题的一种...
Optimal Transport (OT) 是一种数学理论，专注于描述在不同数据分布之间进行最优传输的策略。此理论适用广泛，包括图像处理、机器学习、统计学和计算流体力学等领域。在OT理论中，两个分布被视作“质量”分布，目标是找到最有效方式在它们之间传输数据，同时保持“质量”不变。通过解决费用优化问题实现这一...
We prove that the strategy converges to the solution of the OT problem as the order increases. We also show how to approximate linear quantities of interest, and how to estimate the support of the optimal transport map from the computed moments using Christoffel-Darboux kernels. Numerical ...
The optimal transport (OT) problem is a classical optimization problem having the form of linear programming. Machine learning applications put forward new computational challenges in their solution. In particular, the OT problem defines a distance betwe
In this work, we study the optimal transport (OT) problem between symmetric positive definite (SPD) matrix-valued measures. We formulate the above as a generalized optimal transport problem where the cost, the marginals, and the coupling are represented as block matrices and each component block ...
(OT) problem using Wasserstein-1 distance. However, it was not clear how cycleGAN-type generative models can be derived from the optimal transport theory. Here we show that a novel cycleGAN architecture can be derived as a Kantorovich dual OT formulation if a penalized least square (PLS) cost...
RegOT is a collection of state-of-the-art solvers for regularized optimal transport (OT) problems, implemented in efficient C++ code. This repository is the Python interface to RegOT. 📝 Formulation RegOT mainly solves two types of regularized OT problems: the entropic-regularized OT (EROT) ...