Relative-Translation Invariant Wasserstein Distance

Abstract

We introduce a new family of distances, relative-translation invariant Wasserstein distances (RW_p), for measuring the similarity of two probability distributions under distribution shift. Generalizing it from the classical optimal transport model, we show that RW_p distances are also real distance metrics defined on the quotient set P_p(R^n)/ and invariant to distribution translations. When p=2, the RW_2 distance enjoys more exciting properties, including decomposability of the optimal transport model, translation-invariance of the RW_2 distance, and a Pythagorean relationship between RW_2 and the classical quadratic Wasserstein distance (W_2). Based on these properties, we show that a distribution shift, measured by W_2 distance, can be explained in the bias-variance perspective. In addition, we propose a variant of the Sinkhorn algorithm, named RW_2 Sinkhorn algorithm, for efficiently calculating RW_2 distance, coupling solutions, as well as W_2 distance. We also provide the analysis of numerical stability and time complexity for the proposed algorithm. Finally, we validate the RW_2 distance metric and the algorithm performance with three experiments. We conduct one numerical validation for the RW_2 Sinkhorn algorithm and show two real-world applications demonstrating the effectiveness of using RW_2 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 RW_2 distance is robust to distribution translations compared with baselines.

Tasks

Reproductions