# On the Analysis of EM for truncated mixtures of two Gaussians

arXiv: https://arxiv.org/abs/1902.06958 · HTML: https://mlsanity.com/paper/1902.06958/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Feb 2019.

## Built on by (most cited first, 10 of 10)

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- [Smoothed Analysis of Learning from Positive Samples](https://mlsanity.com/paper/2504.10428/) (2025)
- [EM's Convergence in Gaussian Latent Tree Models](https://mlsanity.com/paper/2211.11904/) (2022)
- [Learning Mixtures of Gaussians with Censored Data](https://mlsanity.com/paper/2305.04127/) (2023)
- [Learning the Inverse Temperature of Ising Models under Hard Constraints using One Sample](https://mlsanity.com/paper/2509.20993/) (2025)
- [Efficient Learning of Truncated Boolean Product Distributions: Influence to the Rescue](https://mlsanity.com/paper/2607.22889/) (2026)

## Abstract (arXiv)

Motivated by a recent result of Daskalakis et al. 2018, we analyze the population version of Expectation-Maximization (EM) algorithm for the case of textittruncated mixtures of two Gaussians. Truncated samples from a d-dimensional mixture of two Gaussians frac12 N(vecμ, vecSigma)+ frac12 N(-vecμ, vecSigma) means that a sample is only revealed if it falls in some subset S subset 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 -vecμ, vec0, vecμ (covariance matrix vecSigma 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).