Relative-Translation Invariant Wasserstein Distance

We introduce a new family of distances, relative-translation invariant Wasserstein distances (RWₚ), for measuring the similarity of two probability distributions under distribution shift. Generalizing it from the classical optimal transport model, we show that RWₚ distances are also real distance metrics defined on the quotient set Pₚ (Rⁿ) / and invariant to distribution translations. When p=2, the RW₂ distance enjoys more exciting properties, including decomposability of the optimal transport model, translation-invariance of the RW₂ distance, and a Pythagorean relationship between RW₂ and the classical quadratic Wasserstein distance (W₂). Based on these properties, we show that a distribution shift, measured by W₂ distance, can be explained in the bias-variance perspective. In addition, we propose a variant of the Sinkhorn algorithm, named RW₂ Sinkhorn algorithm, for efficiently calculating RW₂ distance, coupling solutions, as well as W₂ distance. We also provide the analysis of numerical stability and time complexity for the proposed algorithm. Finally, we validate the RW₂ distance metric and the algorithm performance with three experiments. We conduct one numerical validation for the RW₂ Sinkhorn algorithm and show two real-world applications demonstrating the effectiveness of using RW₂ under distribution shift: digits recognition and similar thunderstorm detection. The experimental results report that our proposed algorithm significantly improves the computational efficiency of Sinkhorn in certain practical applications, and the RW₂ distance is robust to distribution translations compared with baselines.