  The expectation-maximization (EM) algorithm and its variants are widely used
in statistics. In high-dimensional mixture linear regression, the model is
assumed to be a finite mixture of linear regression and the number of
predictors is much larger than the sample size. The standard EM algorithm,
which attempts to find the maximum likelihood estimator, becomes infeasible for
such model. We devise a group lasso penalized EM algorithm and study its
statistical properties. Existing theoretical results of regularized EM
algorithms often rely on dividing the sample into many independent batches and
employing a fresh batch of sample in each iteration of the algorithm. Our
algorithm and theoretical analysis do not require sample-splitting, and can be
extended to multivariate response cases. The proposed methods also have
encouraging performances in numerical studies.
