  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.
