We introduce a modified Benamou-Brenier type approach leading to a Wasserstein type distance that allows global invariance, specifically, isometries, and we show that the problem can be summarized to orthogonal transformations. This distance is defined by penalizing the action with a costless movement of the particle that does not change the direction and speed of its trajectory. We show that for Gaussian distribution resume to measuring the Euclidean distance between their ordered vector of eigenvalues and we show a direct application in recovering Latent Gaussian distributions.

翻译：我们提出了一种改进的贝纳穆-布雷尼耶型方法，由此导出一个允许全局不变性（特别是等距变换）的瓦瑟斯坦型距离，并证明该问题可归结为正交变换。该距离的定义方式是对作用量进行惩罚，允许粒子在不改变其轨迹方向和速度的情况下进行无成本移动。我们证明，对于高斯分布，该距离可归结为测量其有序特征值向量之间的欧几里得距离，并展示了该方法在恢复潜在高斯分布中的直接应用。