In this paper, we study the low-rank matrix optimization problem, where the penalty term is the $\ell_p~(0<p<1)$ regularization. Inspired by the good performance of half thresholding function in sparse/low-rank recovery problems, we propose a singular value half thresholding (SVHT) algorithm to solve the $\ell_p$ regularized matrix optimization problem. The main iteration in SVHT algorithm makes use of the closed-form solution of the subproblem but is different from the existing algorithm, which is in essence to make local 1/2 approximation to the objective function at the current point, instead of local linear or local quadratic approximation. By constructing Lipschitz and non-Lipschitz approximate functions of the objective function, we prove that any accumulation point of the sequence generated by SVHT algorithm is a first-order stationary point of the problem. In numerical experiments, we test SVHT algorithm by low-rank matrix completion problem on both simulated data and real image data. Extensive numerical results show that SVHT algorithm is very efficient for low-rank matrix optimization problems in terms of speed, accuracy, robustness and so on.

## Article

View Singular value half thresholding algorithm for lp regularized matrix optimization problems