  In this paper, we consider high-dimensional nonconvex square-root-loss
regression problems and introduce a proximal majorization-minimization (PMM)
algorithm for these problems. Our key idea for making the proposed PMM to be
efficient is to develop a sparse semismooth Newton method to solve the
corresponding subproblems. By using the Kurdyka-{\L}ojasiewicz property
exhibited in the underlining problems, we prove that the PMM algorithm
converges to a d-stationary point. We also analyze the oracle property of the
initial subproblem used in our algorithm. Extensive numerical experiments are
presented to demonstrate the high efficiency of the proposed PMM algorithm.
