Convergence of Gradient EM on Multi-component Mixture of Gaussians
Convergence of Gradient EM on Multi-component Mixture of Gaussians
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发表时间：
2017
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通讯作者：
Bowei Yan;Mingzhang Yin;Purnamrita Sarkar
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作者：
Bowei Yan;Mingzhang Yin;Purnamrita Sarkar
本文研究了高斯混合模型中期望最大化算法的梯度变量在任意簇个数和混合系数下的收敛性质。我们推导出了收敛速度依赖于混合系数、真实中心之间的最小和最大成对距离、维度和分量数目，并得到了一个接近最优的局部收缩半径。虽然最近已经有一些值得注意的工作来推导出EM在两个对称的高斯混合中的局部收敛速度，但在更一般的情况下，导子需要结构上不同的非平凡参数。我们使用学习理论和实证过程中的最新工具来实现我们的理论结果。
In this paper, we study convergence properties of the gradient variant of Expectation-Maximization algorithm~\cite{lange1995gradient} for Gaussian Mixture Models for arbitrary number of clusters and mixing coefficients. We derive the convergence rate depending on the mixing coefficients, minimum and maximum pairwise distances between the true centers, dimensionality and number of components; and obtain a near-optimal local contraction radius. While there have been some recent notable works that derive local convergence rates for EM in the two symmetric mixture of Gaussians, in the more general case, the derivations need structurally different and non-trivial arguments. We use recent tools from learning theory and empirical processes to achieve our theoretical results.