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).

本文介绍了用期望最大化 (EM) 算法估计两个线性回归混合模型系数的收敛性，根据实验结果，我们发现算法需要在无限圆锥内开始初始化。同时发现，如果初始猜测与目标参数向量的余弦角足够大，则 EM 算法的样本分割版本可以高概率地收敛到真实系数向量。