  Mixed linear regression (MLR) model is among the most exemplary statistical
tools for modeling non-linear distributions using a mixture of linear models.
When the additive noise in MLR model is Gaussian, Expectation-Maximization (EM)
algorithm is a widely-used algorithm for maximum likelihood estimation of MLR
parameters. However, when noise is non-Gaussian, the steps of EM algorithm may
not have closed-form update rules, which makes EM algorithm impractical. In
this work, we study the maximum likelihood estimation of the parameters of MLR
model when the additive noise has non-Gaussian distribution. In particular, we
consider the case that noise has Laplacian distribution and we first show that
unlike the the Gaussian case, the resulting sub-problems of EM algorithm in
this case does not have closed-form update rule, thus preventing us from using
EM in this case. To overcome this issue, we propose a new algorithm based on
combining the alternating direction method of multipliers (ADMM) with EM
algorithm idea. Our numerical experiments show that our method outperforms the
EM algorithm in statistical accuracy and computational time in non-Gaussian
noise case.
