  Novel sparse reconstruction algorithms are proposed for beamspace channel
estimations in massive multiple-input multiple-output systems. The proposed
algorithms minimize a least-squares objective with a nonconvex regularizer.
This regularizer removes the penalties on a few large-magnitude elements from
the conventional l1-norm regularizer, and thus it only forces penalties on the
remaining elements that are expected to be zeros. Accurate and fast
reconstructions can be achieved by performing gradient projection updates in a
difference of convex functions (DC) programming framework. A double-loop
algorithm and a single-loop algorithm are derived by different DC
decompositions, and they have distinct computation complexities and convergence
rates. An extension algorithm is further proposed to generalize the step size
of the single-loop algorithm. This extension algorithm has a faster convergence
rate and can achieve the same level of accuracy as the proposed double-loop
algorithm. Numerical results show significant advantages of the proposed
algorithms over the existing reconstruction algorithms in terms of
reconstruction accuracies and runtimes.
