Estimating the Coefficients of a Mixture of Two Linear Regressions by Expectation Maximization

Jason M. Klusowski, Dana Yang, W. D. Brinda

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


We give convergence guarantees for estimating the coefficients of a symmetric mixture of two linear regressions by expectation maximization (EM). In particular, we show that the empirical EM iterates converge to the target parameter vector at the parametric rate, provided the algorithm is initialized in an unbounded cone. In particular, if the initial guess has a sufficiently large cosine angle with the target parameter vector, a sample-splitting version of the EM algorithm converges to the true coefficient vector with high probability. Interestingly, our analysis borrows from tools used in the problem of estimating the centers of a symmetric mixture of two Gaussians by EM. We also show that the population of EM operator for mixtures of two regressions is anti-contractive from the target parameter vector if the cosine angle between the input vector and the target parameter vector is too small, thereby establishing the necessity of our conic condition. Finally, we give empirical evidence supporting this theoretical observation, which suggests that the sample-based EM algorithm may not converge to the target vector when initial guesses are drawn accordingly. Our simulation study also suggests that the EM algorithm performs well even under model misspecification (i.e., when the covariate and error distributions violate the model assumptions).

Original languageEnglish (US)
Article number8606170
Pages (from-to)3515-3524
Number of pages10
JournalIEEE Transactions on Information Theory
Issue number6
StatePublished - Jun 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


  • Mixture models
  • clustering algorithms
  • expectation-maximization algorithm
  • iterative algorithms
  • regression analysis


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