Elastic Optimal Transport: Theory, Application, and Empirical Evaluation

The classical optimal transport such as Kantorovich's optimal transport and partial optimal transport could be too restrictive in applications due to the full-mass or fixed-mass preservation constraints. To remedy this limitation, we propose elastic optimal transport (ELOT) which is distinctive from the classical optimal transport in its ability of adaptive-mass preserving. It aims to answer the problem of how to transport the probability mass adaptively between probability distributions, which is a fundamental topic in various areas of artificial intelligence. The strength of elastic optimal transport is its capability to transport adaptive-mass in the light of the geometry structure of the problem itself. As an application example in machine learning, we apply elastic optimal transport to both unsupervised domain adaptation and partial domain adaptation tasks. It adaptively transports masses from source domain to target domain by taking domain shift into consideration and respecting the ubiquity of noises or outliers in the data, in order to improve the generalization performance. The experiment results on the benchmarks show that ELOT significantly outperforms the state-of-the-art methods. As a powerful distribution matching tool, elastic optimal transport might be of interests to the broad areas such as artificial intelligence, healthcare, physics, operations research, urban science, etc. The source code is available in the supplementary material.