Reviews: Convergence of Gradient EM on Multi-component Mixture of Gaussians

Summary: This paper derives statistical guarantees for the gradient EM algorithm applied to clustering in Gaussian mixture models, generalizing the results of [1] to multiple clusters and non-uniform cluster weights. The first result (Theorem 1) concerns the "population EM algorithm", i.e., when the expectation step can be computed exactly rather than being estimated from data. For this case, if the cluster centers are sufficiently separated and all the estimated cluster centers are initialized sufficiently close to the true cluster centers (i.e., roughly if there is already a sufficiently clear 1-1 correspondence between the estimated and true cluster centers), then the estimated cluster centers converge linearly to the true centers. This result relies on first showing (Theorem 4) that, under these conditions, due to properties of the Gaussian distribution, the curvature of the Q function is bounded (i.e., specifically, the gradient is Lipschitz). By a standard proof, the Lipschitz gradient then implies convergence of gradient descent.