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.

We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation.

We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric.

R2 considered our method requiring a "discretized proxy." First of all, a different, more challenging optimization problem is studied in our work. The variables in the16 barycenter problem we consider include not only the individual transport plan from each source to the barycenter,17 but importantly also the barycenter itself. Wewould33 like to point out that there are three accepted papers at NeurIPS last year inspired by Wasserstein barycenters. These are37 challenging questions that depend on the specific structure of parameterization and the particular recovery method.38

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.

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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.