[
    {
        "id": "osp-16933",
        "type": "article-journal",
        "title": "Is $\\sqrt{d}$ Separation Necessary for Gradient EM to Learn Gaussian Mixtures in High Dimensions?",
        "author": [
            {
                "family": "Zhang",
                "given": "Yiran"
            },
            {
                "family": "Zhou",
                "given": "Mo"
            },
            {
                "family": "Xu",
                "given": "Weihang"
            },
            {
                "family": "Fazel",
                "given": "Maryam"
            },
            {
                "family": "Du",
                "given": "Simon S."
            }
        ],
        "URL": "https://omanscience.com/en/articles/is-sqrt-d-separation-necessary-for-gradient-em-to-learn-gaussian-mixtures-in-high-dimensions",
        "language": "en",
        "issued": {
            "date-parts": [
                [
                    2026
                ]
            ]
        },
        "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 $Ω(\\sqrt{d})$, 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 $ε> 0$, we prove that when the dimension is sufficiently large, in the worst case a separation of order $Ω(d^{0.5-ε})$ 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."
    }
]