Efficiently aggregating data from different sources is a challenging problem, particularly when samples from each source are distributed differently. These differences can be inherent to the inference task or present for other reasons: sensors in a sensor network may be placed far apart, affecting their individual measurements. Conversely, it is computationally advantageous to split Bayesian inference tasks across subsets of data, but data need not be identically distributed across subsets. One principled way to fuse probability distributions is via the lens of optimal transport: the Wasserstein barycenter is a single distribution that summarizes a collection of input measures while respecting their geometry. However, computing the barycenter scales poorly and requires discretization of all input distributions and the barycenter itself.

While OT commonly seeks at computing the transport plan that minimizes the cost of moving between two distributions, it can naturally be extended to the multi-marginal setting (mOT) when considering several distributions.

Sampling and inference algorithms behave poorly as the number of modes increases, andthisproblem isonlyexacerbated inthiscontextsinceincreasing thenumber ofcomponents in the mixture model leads to a super-exponential increase in the number of modes of the posterior.

Theworddistributions of topics, their optimal transports to the word distributions of documents, and the embeddings of words are learned in a unified framework. When learning thetopic model, weleverage adistilled underlying distance matrix toupdate the topic distributions and smoothly calculate the corresponding optimal transports.

Optimal transport (OT) [30, 25] has become increasingly popular in the machine learning and optimizationcommunity.

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