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How well do SEM algorithms imitate EM algorithms? A non-asymptotic analysis for mixture models

机译:SEM算法如何模仿EM算法? 混合模型的非渐近分析

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In this paper, we present a theoretical and an experimental comparison of EM and SEM algorithms for different mixture models. The SEM algorithm is a stochastic variant of the EM algorithm. The qualitative intuition behind the SEM algorithm is simple: If the number of observations is large enough, then we expect that an update step of the stochastic SEM algorithm is similar to the corresponding update step of the deterministic EM algorithm. In this paper, we quantify this intuition. We show that with high probability the update equations of any EM-like algorithm and its stochastic variant are similar, given that the input set satisfies certain properties. For instance, this result applies to the well-known EM and SEM algorithm for Gaussian mixture models and EM-like and SEM-like heuristics for multivariate power exponential distributions. Our experiments confirm that our theoretical results also hold for a large number of successive update steps. Thereby we complement the known asymptotic results for the SEM algorithm. We also show that, for multivariate Gaussian and multivariate Laplacian mixture models, an update step of SEM runs nearly twice as fast as an EM update set.
机译:在本文中,我们提出了不同混合模型的理论和实验比较EM和SEM算法。 SEM算法是EM算法的随机变体。 SEM算法背后的定性直觉很简单:如果观察的数量足够大,则预期随机SEM算法的更新步骤类似于确定性EM算法的相应更新步骤。在本文中,我们量化了这种直觉。我们表明,具有高概率,鉴于输入组满足某些属性,因此具有高概率和其随机算法的更新方程及其随机变量。例如,该结果适用于可众所周知的EM和SEM算法,用于高斯混合模型和EM样和SEM样启发式,用于多变量功率指数分布。我们的实验证实,我们的理论结果也适用于大量连续更新步骤。因此,我们补充了SEM算法的已知渐近结果。我们还表明,对于多变量高斯和多变量Laplacian混合模型,SEM的更新步骤几乎速度为EM更新集的两倍。

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