Relative Translation Invariant Wasserstein Distance

TMLR Paper6929 Authors

Abstract:

Motivated by the Bures-Wasserstein distance, we introduce a new family of \emph{relative translation invariant Wasserstein distances}, denoted $(RW_p)$, 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 robust to perturbation than the classical Wasserstein distances. A bi-level algorithm is designed to compute the general $RW_p$ distances between arbitrary discrete distributions.
Additionally, when $p = 2$, we show that the optimal coupling solutions are invariant under distributional translation in discrete settings, and we further propose two algorithms, the $\mathrm{RW}_2$-Sinkhorn algorithm and $\mathrm{RW}_2$-LP algorithm, to improve the numerical stability of computing $W_2$ distances and the optimal coupling solutions.
Finally, we conduct three experiments to validate our theoretical results and algorithms. The first two experiments report that the $\mathrm{RW}_2$-Sinkhorn algorithm and $\mathrm{RW}_2$-LP algorithm 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.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Rémi_Flamary1

Submission Number: 6929