Optimal transport (OT) is a powerful tool for measuring the distance between two probability distributions. In this paper, we introduce a new manifold named as the coupling matrix manifold (CMM), where each point on this novel manifold can be regarded as a transportation plan of the optimal transport problem. We firstly explore the Riemannian geometry of CMM with the metric expressed by the Fisher information. These geometrical features can be exploited in many essential optimization methods as a framework solving all types of OT problems via incorporating numerical Riemannian optimization algorithms such as gradient descent and trust region algorithms in CMM manifold. The proposed approach is validated using several OT problems in comparison with recent state-of-the-art related works. For the classic OT problem and its entropy regularized variant, it is shown that our method is comparable with the classic algorithms such as linear programming and Sinkhorn algorithms. For other types of non-entropy regularized OT problems, our proposed method has shown superior performance to other works, whereby the geometric information of the OT feasible space was not incorporated within.

中文翻译：

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耦合矩阵流形辅助优化优化传输问题

最优输运 (OT) 是测量两个概率分布之间距离的强大工具。在本文中，我们引入了一种称为耦合矩阵流形（CMM）的新流形，其中这个新流形上的每个点都可以看作是最优运输问题的运输计划。我们首先用Fisher信息表示的度量来探索CMM的黎曼几何。通过在 CMM 流形中结合数值黎曼优化算法（例如梯度下降和信任区域算法），这些几何特征可以在许多基本优化方法中作为解决所有类型 OT 问题的框架加以利用。与最近最先进的相关工作相比，所提出的方法使用几个 OT 问题进行了验证。对于经典的 OT 问题及其熵正则化变体，结果表明，我们的方法可与线性规划和 Sinkhorn 算法等经典算法相媲美。对于其他类型的非熵正则化 OT 问题，我们提出的方法表现出优于其他工作的性能，其中不包含 OT 可行空间的几何信息。