Is sqrt{d} Separation Necessary for Gradient EM to Learn Gaussian Mixtures in High Dimensions?

arXiv:2610.07551v1 Announce Type: cross Abstract: Learning Gaussian mixture models (GMMs) using the Expectation-Maximization (EM) algorithm and its gradient-based variants is a fundamental problem in machine learning. It is known that randomly initialized (gradient) EM fails to learn multi-component GMMs in the exact-parameterized setting, where the number of components matches that of the ground-truth GMM. Recently, global convergence of gradient EM has been established in the over-parameterized setting, where more components are used, provided that the ground-truth components are well separated. In particular, the minimum separation between ground-truth components is required to scale as Omega(sqrt{d}), where d is the dimension. In this paper, we show that this dimensional dependence is unavoidable in high-dimensional settings. Specifically, we consider a hybrid EM algorithm that uses standard EM updates for the mixing weights and gradient EM updates for the component means. For any epsilon > 0, we prove that when the dimension is sufficiently large, in the worst case a separation of order Omega(d^{0.5-epsilon}) is insufficient to guarantee global convergence of population gradient EM in sub-exponential time under random initialization, even in the over-parameterized regime. Our result establishes an almost optimal worst-case lower bound on the ground-truth separation required for learning Gaussian mixtures via gradient EM in high dimensions.

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Source: arXiv cs.LG | 2026-10-07