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Abstract:
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In the optimal transport problem, one seeks to minimize the expected value of a bivariate cost function over joint distributions with fixed marginals. Recently, optimal transport based techniques have been successfully applied to a number of generative modeling and supervised learning tasks. We describe a theoretical framework, and accompanying computational tools, that extends optimal transport ideas to stationary Markov chains and hidden Markov models. The framework, which is built on the notion of transition couplings, provides a means of comparing and synchronizing Markov chains and HMMs. We motivate and illustrate the framework with examples of graph alignment and synchronization of computer generated music.
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