  Mixtures of Linear Regressions (MLR) is an important mixture model with many
applications. In this model, each observation is generated from one of the
several unknown linear regression components, where the identity of the
generated component is also unknown. Previous works either assume strong
assumptions on the data distribution or have high complexity. This paper
proposes a fixed parameter tractable algorithm for the problem under general
conditions, which achieves global convergence and the sample complexity scales
nearly linearly in the dimension. In particular, different from previous works
that require the data to be from the standard Gaussian, the algorithm allows
the data from Gaussians with different covariances. When the conditional number
of the covariances and the number of components are fixed, the algorithm has
nearly optimal sample complexity $N = \tilde{O}(d)$ as well as nearly optimal
computational complexity $\tilde{O}(Nd)$, where $d$ is the dimension of the
data space. To the best of our knowledge, this approach provides the first such
recovery guarantee for this general setting.
