Convergence of Gradient EM on Multi-component Mixture of Gaussians

In this paper, we study convergence properties of the gradient variant of Expectation-Maximization algorithm [11] for Gaussian Mixture Models for arbitrary numberof clusters and mixing coefficients. We derive the convergence rate depending on the mixing coefficients, minimum and maximum pairwise distances betweenthe 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 nontrivial arguments. We use recent tools from learning theory and empirical processes to achieve our theoretical results.