On the Analysis of EM for truncated mixtures of two Gaussians
This addresses the problem of parameter estimation in truncated Gaussian mixtures for statisticians and machine learning researchers, offering theoretical guarantees for EM convergence in a challenging scenario, though it is incremental as it builds on prior work on untruncated mixtures.
The paper analyzes the population version of the Expectation-Maximization (EM) algorithm for truncated mixtures of two Gaussians, showing that for one-dimensional cases, EM converges almost surely to the true mean for any measurable truncation set, and for higher dimensions, it converges under conditions of three fixed points, with local convergence otherwise, and provides convergence rates.
Motivated by a recent result of Daskalakis et al. 2018, we analyze the population version of Expectation-Maximization (EM) algorithm for the case of \textit{truncated} mixtures of two Gaussians. Truncated samples from a $d$-dimensional mixture of two Gaussians $\frac{1}{2} \mathcal{N}(\vecμ, \vecΣ)+ \frac{1}{2} \mathcal{N}(-\vecμ, \vecΣ)$ means that a sample is only revealed if it falls in some subset $S \subset \mathbb{R}^d$ of positive (Lebesgue) measure. We show that for $d=1$, EM converges almost surely (under random initialization) to the true mean (variance $σ^2$ 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 $-\vecμ, \vec{0}, \vecμ$ (covariance matrix $\vecΣ$ 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).