Relative-Translation Invariant Wasserstein Distance

• URL: http://arxiv.org/abs/2409.02416v1
• Date: Wed, 4 Sep 2024 03:41:44 GMT
• Title: Relative-Translation Invariant Wasserstein Distance
• Authors: Binshuai Wang, Qiwei Di, Ming Yin, Mengdi Wang, Quanquan Gu, Peng Wei,
• Abstract summary: We introduce a new family of distances, relative-translation invariant Wasserstein distances ($RW_p$) We show that $RW_p distances are also real distance metrics defined on the quotient set $mathcalP_p(mathbbRn)/sim$ invariant to distribution translations.
• Score: 82.6068808353647
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\mathcal{P}_p(\mathbb{R}^n)/\sim$ 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.

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