Relative Translation Invariant Wasserstein Distance

arXiv:2409.02416v2 Announce Type: replace Abstract: Motivated by the Bures distance, we introduce a new family of distances, 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 RW_2-LP algorithm and the 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 RW_2-LP algorithm and the 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.

Source: arXiv cs.LG | 2026-05-26