  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 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 performs poorly 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).
