论文标题
通用的非标准化最佳运输及其快速算法
Generalized Unnormalized Optimal Transport and its fast algorithms
论文作者
论文摘要
我们引入了快速算法，用于通用的非均衡最佳运输。为了处理总质量不同的密度，我们考虑了一个动态模型，该模型将$ l^p $最佳传输与$ l^p $距离混合在一起。对于$ p = 1 $，我们得出相应的$ l^1 $概括的kantorovich公式。我们进一步表明，问题变成了一个简单的$ l^1 $最小化，该算法有效地解决了。对于$ p = 2 $，我们得出了$ l^2 $概括的kantorovich公式，一个新的非标准蒙格问题和相应的monge-ampère方程。此外，我们介绍了问题的新不受约束的优化公式。相关的梯度流与可以有效求解的椭圆方程基本有关。在这里，提出的梯度下降程序以及Nesterov加速度涉及由KKT条件引起的汉密尔顿 - 雅各比方程。提出了几个数值示例，以说明所提出的算法的有效性。
We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the $L^p$ optimal transport with $L^p$ distance. For $p=1$, we derive the corresponding $L^1$ generalized unnormalized Kantorovich formula. We further show that the problem becomes a simple $L^1$ minimization which is solved efficiently by a primal-dual algorithm. For $p=2$, we derive the $L^2$ generalized unnormalized Kantorovich formula, a new unnormalized Monge problem and the corresponding Monge-Ampère equation. Furthermore, we introduce a new unconstrained optimization formulation of the problem. The associated gradient flow is essentially related to an elliptic equation which can be solved efficiently. Here the proposed gradient descent procedure together with the Nesterov acceleration involves the Hamilton-Jacobi equation which arises from the KKT conditions. Several numerical examples are presented to illustrate the effectiveness of the proposed algorithms.
