  Novel sparse reconstruction algorithms are proposed for beamspace channel
estimation in massive multiple-input multiple-output systems. The proposed
algorithms minimize a least-squares objective having 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
within the framework of difference of convex functions (DC) programming. A
double-loop algorithm and a single-loop algorithm are proposed via different DC
decompositions, and these two algorithms have distinct computation complexities
and convergence rates. Then, an extension algorithm is further proposed by
designing the step sizes of the single-loop algorithm. The extension algorithm
has a faster convergence rate and can achieve approximately the same level of
accuracy as the proposed double-loop algorithm. Numerical results show
significant advantages of the proposed algorithms over existing reconstruction
algorithms in terms of reconstruction accuracies and runtimes. Compared to the
benchmark channel estimation techniques, the proposed algorithms also achieve
smaller mean squared error and higher achievable spectral efficiency.
