  We consider the problem of solving mixed random linear equations with $k$
components. This is the noiseless setting of mixed linear regression. The goal
is to estimate multiple linear models from mixed samples in the case where the
labels (which sample corresponds to which model) are not observed. We give a
tractable algorithm for the mixed linear equation problem, and show that under
some technical conditions, our algorithm is guaranteed to solve the problem
exactly with sample complexity linear in the dimension, and polynomial in $k$,
the number of components. Previous approaches have required either exponential
dependence on $k$, or super-linear dependence on the dimension. The proposed
algorithm is a combination of tensor decomposition and alternating
minimization. Our analysis involves proving that the initialization provided by
the tensor method allows alternating minimization, which is equivalent to EM in
our setting, to converge to the global optimum at a linear rate.
