  In this paper, we focus on the local convergence rate analysis of the
proximal iteratively reweighted $\ell_1$ algorithms for solving $\ell_p$
regularization problems, which are widely applied for inducing sparse
solutions. We show that if the Kurdyka-Lojasiewicz (KL) property is satisfied,
the algorithm converges to a unique first-order stationary point; furthermore,
the algorithm has local linear convergence or local sublinear convergence. The
theoretical results we derived are much stronger than the existing results for
iteratively reweighted $\ell_1$ algorithms.
