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

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 random 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, for any $\epsilon > 0$, we prove that when the dimension is sufficiently large, 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 lower bound on the ground-truth separation required for learning Gaussian mixtures via gradient EM in high dimensions.