We study the trajectory of iterations and the convergence rates of the
Expectation-Maximization (EM) algorithm for two-component Mixed Linear
Regression (2MLR). The fundamental goal of MLR is to learn the regression
models from unlabeled observations. The EM algorithm finds extensive
applications in solving the mixture of linear regressions. Recent results have
established the super-linear convergence of EM for 2MLR in the noiseless and
high SNR settings under some assumptions and its global convergence rate with
random initialization has been affirmed. However, the exponent of convergence
has not been theoretically estimated and the geometric properties of the
trajectory of EM iterations are not well-understood. In this paper, first,
using Bessel functions we provide explicit closed-form expressions for the EM
updates under all SNR regimes. Then, in the noiseless setting, we completely
characterize the behavior of EM iterations by deriving a recurrence relation at
the population level and notably show that all the iterations lie on a certain
cycloid. Based on this new trajectory-based analysis, we exhibit the
theoretical estimate for the exponent of super-linear convergence and further
improve the statistical error bound at the finite-sample level. Our analysis
provides a new framework for studying the behavior of EM for Mixed Linear
Regression.

我们研究了两组分混合线性回归 (2MLR) 的期望最大化 (EM) 算法的迭代轨迹和收敛速度。通过使用贝塞尔函数，我们提供了 EM 在所有信噪比情况下的明确闭式表达式，并在无噪音的情况下通过导出一个循环关系完全刻画了 EM 迭代的行为，从而揭示了所有迭代都位于某个摆线上。基于这种新的轨迹分析，我们展示了超线性收敛指数的理论估计，并进一步提高了有限样本层面上的统计误差界限。我们的分析为研究混合线性回归中的 EM 行为提供了新的框架。