Title: On the contraction properties of Sinkhorn semigroups Show Chinese title

Title: 关于Sinkhorn半群的收缩性质

Abstract: We develop a novel semigroup contraction analysis based on Lyapunov techniques to prove the exponential convergence of Sinkhorn equations on weighted Banach spaces. This operator-theoretic framework yields exponential decays of Sinkhorn iterates towards Schr\"odinger bridges with respect to general classes of $\phi$-divergences as well as in weighted Banach spaces. To the best of our knowledge, these are the first results of this type in the literature on entropic transport and the Sinkhorn algorithm. We also illustrate the impact of these results in the context of multivariate linear Gaussian models as well as statistical finite mixture models including Gaussian-kernel density estimation of complex data distributions arising in generative models. Show Chinese abstract

Abstract: 我们开发了一种基于Lyapunov技术的新颖半群收缩分析，以证明Sinkhorn方程在加权Banach空间上的指数收敛性。 该算子理论框架在一般类别的$\phi$-散度以及加权Banach空间中，产生了Sinkhorn迭代 toward Schrödinger桥的指数衰减。 据我们所知，这些是关于熵运输和Sinkhorn算法文献中首次出现的此类结果。 我们还展示了这些结果在多元线性高斯模型以及统计有限混合模型中的影响，包括生成模型中出现的复杂数据分布的高斯核密度估计。