Selecting Regularization Parameters for Nuclear Norm--Type Minimization Problems

Abstract

The reconstruction of a low-rank matrix from its noisy observation finds its usage in many applications. It can be reformulated into a constrained nuclear norm minimization problem, where the bound $\eta$ of the constraint is explicitly given or can be estimated by the probability distribution of the noise. When the Lagrangian method is applied to find the minimizer, the solution can be obtained by the singular value thresholding operator, where the thresholding parameter $\lambda$ is related to the Lagrangian multiplier. In this paper, we first show that the Frobenius norm of the discrepancy between the minimizer and the observed matrix is a strictly monotonically increasing function of $\lambda$. From that we derive a closed form solution for $\lambda$ in terms of $\eta$. Since $\lambda$ is the same as the regularization parameter for the unconstrained regularized problem, our results can be applied to automatically choosing a suitable regularization parameter for the nuclear norm--type regularized minimization problems using the discrepancy principle. The regularization parameters obtained are comparable to (and sometimes better than) those obtained by Stein's unbiased risk estimator approach, while the cost of solving the minimization problem can be reduced by 11--18 times. Numerical experiments with both synthetic data and real MRI data are performed to validate the proposed approach.