By utilizing recently developed tools for constructing gradient flows on Wasserstein spaces, we extend an analysis technique commonly employed to understand alternating minimization algorithms on Euclidean space to the Expectation Maximization (EM) algorithm via its representation as coordinate-wise minimization on the product of a Euclidean space and a space of probability distributions due to Neal and Hinton (1998). In so doing we obtain finite sample error bounds and exponential convergence of the EM algorithm under a natural generalisation of a log-Sobolev inequality. We further demonstrate that the analysis technique is sufficiently flexible to allow also the analysis of several variants of the EM algorithm.

翻译：通过利用最近开发的在Wasserstein空间上构建梯度流的工具，我们将一种常用于分析欧几里得空间上交替最小化算法的技术，基于Neal和Hinton（1998）提出的将期望最大化（EM）算法表示为欧几里得空间与概率分布空间的笛卡尔积上的坐标最小化方法，扩展到了对EM算法的分析。通过这种方式，我们在对数索博列夫不等式的自然推广下，获得了EM算法的有限样本误差界和指数收敛性。我们进一步证明，该分析技术具有足够的灵活性，也允许对EM算法的若干变体进行分析。