We consider a symmetric mixture of linear regressions with random samples from the pairwise comparison design, which can be seen as a noisy version of a type of Euclidean distance geometry problem. We analyze the expectation-maximization (EM) algorithm locally around the ground truth and establish that the sequence converges linearly, providing an $\ell_\infty$-norm guarantee on the estimation error of the iterates. Furthermore, we show that the limit of the EM sequence achieves the sharp rate of estimation in the $\ell_2$-norm, matching the information-theoretically optimal constant. We also argue through simulation that convergence from a random initialization is much more delicate in this setting, and does not appear to occur in general. Our results show that the EM algorithm can exhibit several unique behaviors when the covariate distribution is suitably structured.

翻译：我们考虑一种对称线性回归混合模型，其随机样本来自成对比较设计，这可视为一类欧几里得距离几何问题的含噪版本。我们围绕真实参数对期望最大化（EM）算法进行局部解析，证明该序列线性收敛，并为迭代估计误差提供了ℓ∞范数保证。进一步地，我们证明EM序列的极限在ℓ2范数下达到估计的精确速率，与信息论最优常数相匹配。通过仿真我们还论证，在此场景下从随机初始化出发的收敛性更为微妙，且通常不会发生。我们的结果表明，当协变量分布具有适当结构时，EM算法会呈现出若干独特行为。