Motivated by the Bures distance, we introduce a new family of distances, \emph{relative translation invariant Wasserstein distances}, denoted by $RW_p$, as an extension of the classical Wasserstein distances $W_p$ for $p \in [1, +\infty)$. We establish that $RW_p$ defines a valid metric and demonstrate that this type of metric is more intrinsic than the classical Wasserstein distance. A bi-level algorithm is designed to compute the general $RW_p$ distance between arbitrary discrete distributions. Moreover, when $p = 2$, we show that the optimal coupling matrix is invariant under distributional translation in the discrete setting, and we further propose two algorithms, the $\mathrm{RW}_2$-LP algorithm and the $\mathrm{RW}_2$-Sinkhorn algorithm, to improve the numerical stability of computing $W_2$ distance and the optimal coupling matrix solutions. Finally, we conduct three experiments to validate our theoretical results and algorithms. The first two experiments report that the $\mathrm{RW}_2$-LP algorithm and the $\mathrm{RW}_2$-Sinkhorn algorithm, both with and without normalization, can significantly reduce the numerical errors compared to standard algorithms. The third experiment shows that $RW_p$ algorithms are computationally scalable and applicable to the retrieval of similar thunderstorm patterns in practical applications.

翻译：受Bures距离的启发，我们引入了一类新的距离族——\emph{相对平移不变Wasserstein距离}（记作$RW_p$），作为经典Wasserstein距离$W_p$（$p \in [1, +\infty)$）的推广。我们证明了$RW_p$定义了一个有效的度量，并表明该度量比经典Wasserstein距离更具内在性。针对任意离散分布间的$RW_p$距离计算，我们设计了一种双层算法。此外，当$p=2$时，我们证明了在离散设定下最优耦合矩阵具有分布平移不变性，并进一步提出了$\mathrm{RW}_2$-LP算法和$\mathrm{RW}_2$-Sinkhorn算法，以提升$W_2$距离及最优耦合矩阵求解的数值稳定性。最后，我们通过三个实验验证了理论结果与算法：前两个实验表明，无论是否进行归一化，$\mathrm{RW}_2$-LP算法与$\mathrm{RW}_2$-Sinkhorn算法均能显著降低数值误差；第三个实验证明$RW_p$算法具有计算可扩展性，可应用于实际场景中相似雷暴模式的检索。