Demystifying EM: New Trajectory-Based Convergence Guarantees for Mixed Linear Regression
By Zhankun Luo, Abolfazl Hashemi
Published on November 10, 2025| Vol. 1, Issue No. 1
Content Source
This is a curated briefing. The original article was published on cs.LG updates on arXiv.org.
Summary\
This work significantly advances the understanding of the Expectation-Maximization (EM) algorithm for two-component Mixed Linear Regression (2MLR), particularly in scenarios where both mixing weights and regression parameters are fully unknown. The authors derive explicit EM update expressions, revealing novel "cycloid trajectories" for its parameters in noiseless and high signal-to-noise ratio (SNR) regimes. Crucially, the study establishes non-asymptotic convergence guarantees, detailing linear or quadratic convergence rates based on the estimate's alignment, and proves robust convergence even with arbitrary initialization. This introduces a powerful new trajectory-based framework for analyzing EM's complex behavior.
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Why It Matters\
The EM algorithm is a cornerstone of statistical machine learning, powering solutions for clustering, missing data imputation, and semi-supervised learning. This paper significantly elevates its practical utility and trustworthiness by addressing the critical gap of its behavior in "fully unknown" Mixed Linear Regression scenarios - the norm in real-world applications where ground truth parameters are elusive. The derivation of non-asymptotic guarantees is a game-changer for practitioners, shifting from theoretical "infinite data" assurances to concrete performance bounds for the finite datasets always encountered in practice. Knowing the convergence rates (linear or quadratic) provides vital insights for optimizing computational resources and predicting algorithm efficiency. Perhaps most impactful is the proof of convergence with arbitrary initialization, which liberates AI professionals from the often-frustrating and heuristic process of finding good starting points, making EM-based solutions for modeling heterogeneous data far more robust and accessible. The novel "cycloid trajectory" framework also offers a deeper analytical lens, potentially inspiring new diagnostic and optimization techniques for a broader class of iterative algorithms, ultimately enhancing the reliability and deployability of foundational AI techniques in complex, real-world problems.