Multi-marginal optimal transport and probabilistic graphical models
This work provides a theoretical extension and computational improvement for multi-marginal optimal transport, benefiting researchers in optimization and machine learning, though it appears incremental.
The paper tackles the multi-marginal optimal transport problem by connecting it to probabilistic graphical models, showing that entropy-regularized transport is equivalent to Bayesian marginal inference with specified marginals, and develops fast algorithms leveraging Bayesian inference methods.
We study multi-marginal optimal transport problems from a probabilistic graphical model perspective. We point out an elegant connection between the two when the underlying cost for optimal transport allows a graph structure. In particular, an entropy regularized multi-marginal optimal transport is equivalent to a Bayesian marginal inference problem for probabilistic graphical models with the additional requirement that some of the marginal distributions are specified. This relation on the one hand extends the optimal transport as well as the probabilistic graphical model theories, and on the other hand leads to fast algorithms for multi-marginal optimal transport by leveraging the well-developed algorithms in Bayesian inference. Several numerical examples are provided to highlight the results.