Wasserstein barycenter, built on the theory of optimal transport, provides a powerful framework to aggregate probability distributions, and it has increasingly attracted great attention within the machine learning community. However, it suffers from severe computational burden, especially for high dimensional and continuous settings. To this end, we develop a novel continuous approximation method for the Wasserstein barycenters problem given sample access to the input distributions. The basic idea is to introduce a variational distribution as the approximation of the true continuous barycenter, so as to frame the barycenters computation problem as an optimization problem, where parameters of the variational distribution adjust the proxy distribution to be similar to the barycenter. Leveraging the variational distribution, we construct a tractable dual formulation for the regularized Wasserstein barycenter problem with c-cyclical monotonicity, which can be efficiently solved by stochastic optimization. We provide theoretical analysis on convergence and demonstrate the practical effectiveness of our method on real applications of subset posterior aggregation and synthetic data.

翻译：瓦塞斯坦温中枢以最佳运输理论为基础,为综合概率分布提供了强大的框架,并日益吸引了机器学习界的极大关注。然而,它承受着严重的计算负担,特别是高维和连续设置。为此,我们为瓦塞斯坦中枢问题开发了一种新的连续近似方法,通过样本访问输入分布,对瓦塞斯坦中枢问题进行了新的连续近似方法。基本想法是引入一种变异分布方法,作为真正连续的中枢的近似,从而将巴塞中心计算问题作为一个优化问题,在这个问题上,变异分布参数将代理分布调整成与中枢相似。我们利用变异分布,我们为正常的瓦塞斯坦中枢问题制造了一种可移植的双向配方,这种配方可以通过随机优化有效解决。我们提供了对趋同的理论分析,并展示了我们在子子子子集和合成数据的实际应用上的方法的实际效力。