2019/02/19 by Sai Ganesh Nagarajan, Ioannis Panageas, Nagarajan, Sai Ganesh +1 · 2 citations
Computer Science · Engineering · Mathematics · #Bayesian Methods and Mixture Models #Control Systems and Identification #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Statistical Distribution Estimation and Applications
paper · pdf · doi:10.48550/arxiv.1902.06958
openalex publication_date 2019/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by a recent result of Daskalakis et al. 2018, we analyze the population version of Expectation-Maximization (EM) algorithm for the case of truncated mixtures of two Gaussians. Truncated samples from a d-dimensional mixture of two Gaussians (1)/(2) N(μ, Σ)+ (1)/(2) N(-μ, Σ) means that a sample is only revealed if it falls in some subset S ⊂ ℝd of positive (Lebesgue) measure. We show that for d=1, EM converges almost surely (under random initialization) to the true mean (variance σ2 is known) for any measurable set S. Moreover, for d>1 we show EM almost surely converges to the true mean for any measurable set S when the map of EM has only three fixed points, namely -μ, 0, μ (covariance matrix Σ is known), and prove local convergence if there are more than three fixed points. We also provide convergence rates of our findings. Our techniques deviate from those of Daskalakis et al. 2017, which heavily depend on symmetry that the untruncated problem exhibits. For example, for an arbitrary measurable set S, it is impossible to compute a closed form of the update rule of EM. Moreover, arbitrarily truncating the mixture, induces further correlations among the variables. We circumvent these challenges by using techniques from dynamical systems, probability and statistics; implicit function theorem, stability analysis around the fixed points of the update rule of EM and correlation inequalities (FKG).