  The low-rank matrix reconstruction (LRMR) approach is widely used in
direction-of-arrival (DOA) estimation. As the rank norm penalty in an LRMR is
NP-hard to compute, the nuclear norm (or the trace norm for a positive
semidefinite (PSD) matrix) has been often employed as a convex relaxation of
the rank norm. However, solving a nuclear norm convex problem may lead to a
suboptimal solution of the original rank norm problem. In this paper, we
propose to apply a family of nonconvex penalties on the singular values of the
covariance matrix as the sparsity metrics to approximate the rank norm. In
particular, we formulate a nonconvex minimization problem and solve it by using
a locally convergent iterative reweighted strategy in order to enhance the
sparsity and resolution. The problem in each iteration is convex and hence can
be solved by using the optimization toolbox. Convergence analysis shows that
the new method is able to obtain a suboptimal solution. The connection between
the proposed method and the sparse signal reconstruction (SSR) is explored
showing that our method can be regarded as a sparsity-based method with the
number of sampling grids approaching infinity. Two feasible implementation
algorithms that are based on solving a duality problem and deducing a
closed-form solution of the simplified problem are also provided for the convex
problem at each iteration to expedite the convergence. Extensive simulation
studies are conducted to show the superiority of the proposed methods.
