Abstract: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.
| Comments: | Accepted by Transactions on Machine Learning Research (TMLR). Final accepted version. The implementation is publicly available at \url{this https URL} |
| Subjects: | Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2409.02416 [cs.LG] |
| (or arXiv:2409.02416v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2409.02416 arXiv-issued DOI via DataCite |
From: Binshuai Wang [view email]
[v1]
Wed, 4 Sep 2024 03:41:44 UTC (2,971 KB)
[v2]
Mon, 25 May 2026 16:44:25 UTC (1,302 KB)
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