Relative Translation Invariant Wasserstein Distance

Motivated by the Bures distance, we introduce a new family of distances, relative translation invariant Wasserstein distances, denoted by RWp, as an extension of the classical Wasserstein distances Wp for p ∈ [1, +∞). We establish that RWp 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 RWp 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 RW2-LP algorithm and the RW2-Sinkhorn algorithm, to improve the numerical stability of computing W2 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 RW2-LP algorithm and the RW2-Sinkhorn algorithm, both with and without normalization, can significantly reduce the numerical errors compared to standard algorithms. The third experiment shows that RWp algorithms are computationally scalable and applicable to the retrieval of similar thunderstorm patterns in practical applications.